Discrete and Continuous Boundary Problems
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Discrete and Continuous Boundary Problems
MATHEMATICS IN SCIENCE A N D E N G I N E E R I N G A Series o f Monographs and Textbooks
Edited by Richard Bellman The RAND Corporation, Santa Monica, California
Volume 1 .
TRACY Y.THOMAS. Concepts from Tensor Analysis and Differential Geometry. 1961
Volume 2.
TRACY Y.THOMAS. Plastic Flow and Fracture in Solids. 1961
Volume 3.
RUTHERFORD ARIS.The Optimal Design of Chemical Reactors: A Study in Dynamic Programming. 1961
Volume 4 .
JOSEPHL SALLEand SOLOMONLEFSCHETZ.Stability by Liapunov's Direct Method with Applications. 1961
Volume 5 .
GEORGELEITMANN(ed.) . Optimization Techniques: with Applications to Aerospace Systems. 1962
Volume 6.
RICHARD BELLMAN and KENNETHL. COOKE.DifferentialDifference Equations. 1963
Volume 7 .
FRANKA. HAIGHT.Mathematical Theories of Traffic Flow. 1963
Vo'olume8 .
F. V. ATKINSON.Discrete and Continuous Boundary Problems. 1964
Volume 9.
A. JEFFREY and T. TANIUTI. Non-Linear Wave Propagation: with Applications to Physics and Magnetohydrodynamics. 1964
Volume 10.
JULIUSTOU.Optimum Design of Digital Control Systems. 1963
Volume 11.
HARLEYFLANDERS. Differential Forms: with Applications to the Physical Sciences. 1963
Volume 12.
SANFORD M. ROBERTS. Dynamic Programming in Chemical Engineering and Process Control. 1964
In preparation D. N. CHORAFAS. Systems and Simulation
Discrete and Continuous B oundury ProbZems F. V. Atkinson DEPARTMENT OF MATHEMATICS UNIVERSITY OF TORONTO TORONTO, CANADA
1964
hTew
York
ACADEMIC PRESS
London
COPYRIGHT @
1964, BY ACADEMIC PRESS INC.
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ACADEMIC PRESS INC. 1 1 1 Fifth Avenue, New York 3, New York
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PRINTED IN THE UNITED STATES OF AMERICA
Preface
A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. [E. T. Bell, “Men of Mathematics,” pp:13-14.
Dover, New York, 1937; Simon and Schuster, New York.]
To compare the discrete with the continuous, to search for analogies between them, and ultimately to effect their unification, are patterns of mathematical development that did not begin with Zeno, and certainly did not end with Leibnitz and Newton, nor even with Riemann and Stieltjes. Such a pattern of investigation is especially appropriate to the theory of boundary problems, for which the discrete and the con$nuous pervade both physical origins and mathematical methods. It is the aim of this book to present in this light the theory of boundary problems in one dimension, that is to say, for “ordinary” differential equations and their analogs and extensions. It would be highly desirable to develop a corresponding theory for partial differential equations and their analogs ; however, here the discrete theory seems ill-developed, and unification remote indeed. The essential unity of our subject has not always been apparent; the wealth of its applications and interpretations are perhaps responsible for this. It has been natural to expound the topic of boundary problems for ordinary differential equations in the situation that the coefficients have any requisite degree of smoothness; this case combines practical value with mathematical convenience. At the other extreme, boundary problems for difference or recurrence relations have tended to be viewed primarily as numerical aids to problems for differential equations. This is not to say that the mathematical theory of recurrence relations has gone undeveloped; the higher branches of the theory of boundary problems for recurrence relations tend to be found more or less effectively disguised in the contexts of linear operators, of the theory of moments, or of continued fractions or of orthogonal polynomials. Here again the dual role of the classical polynomials, as solutions both of differential and of recurrence relations, scarcely lessens the confusion. Unified theories of differential and of difference equations are of too recent emergence to gain full recognition. V
vi
PREFACE
We shall pursue our task from three directions. We shall present the theory of certain recurrence relations in the spirit of the theory of boundary problems for differential equations. Second, we shall present the theory of boundary problems for certain ordinary differential equations, emphasizing cases in which the coefficients may be discontinuous, or may have singularities of delta-function type. Finally, we give some account of theories which unify the topics of differential and difference equations, relying mainly on the method of replacement by integral equations. The introductory Chapter 0 provides a survey of the field to be investigated, and introduces the basic concept for classifying our boundary problems, whether discrete or continuous. This is the invariance of a quadratic form under linear or fractional-linear transformations, which may be continuously or discretely applied; this notion generalizes that of the constancy of the Wronskian in the case of Sturm-Liouville theory. Chapters 1 and 2 take up this notion in the simplest case of the invariance of the modulus of a complex number, to yield what is perhaps the most elementary of boundary problems. Here one may view in microcosm all aspects of the theory; one may also regard the material of these chapters as a primitive case of the as yet relatively undeveloped topic of boundary problems ,involving fractional-linear matrix factors. After Chapter 3, devoted to general principles for recurrence relations, there are four chapters devoted to the spectral theory for special types of recurrence relation. Chapters 4 and 5 might have been entitled “discrete Sturm-Liouville theory,” but many of the results are more familiar in the context of orthogonal polynomials. Chapter 6 presents some recent extensions in the matrix or multi-variate direction. Chapter 7, though its title also relates to orthogonal polynomials, continues to emphasize the recurrence relation approach. With Chapters 8 and 9 we turn to the second aspect of our task, the presentation of the theory of boundary problems for differentialequations, without making unnecessary smoothness restrictions on the coefficients; as various authors have noted, it is in fact possible to arrange for difference equations to be included as special case of differential equations. Chapter 8 deals in this manner with the main case of classical SturmLiouville theory. To conduct a similar investigation for higher-order equations seemed unnecessary here, and, accordingly, Chapter 9 has been confined to an account of the first-order matrix system; this is, of course, more general than the nth order equation and does not seem to have been treated very often in book form. Oscillatory properties for matrix systems have received renewed attention lately, and we have thus devoted Chapter 10 separately to them. Since a number of variational treatments of this topic are available, it
PREFACE
vii
seemed appropriate to expound here the matrix approach. For simplicity, the exposition has been confined to the continuous case. Chapters 11 and 12 are devoted to the unified theory, which includes both differential and recurrence relations. Chapter 11 is devoted to a sketch of the general theory, which has been the object of much recent research, while Chapter 12 deals with the extension of special SturmLiouville properties. Here it must be emphasized that the theory so generalized does not cover the case of fractional-linear relations considered in Chapters I and 2; on the other hand, the theory points the way to numerous generalizations of other investigations for differential equations, apart from boundary problems. The level of mathematical argument is fairly elementary; a knowledge of Lebesgue integration is only rarely needed, while the Stieltjes ihtegral and some of its less accessible properties have been treated in an Appendix. In certain chapters a familiarity with matrix manipulations is presumed ; monotonic properties of eigenvalues have been developed in an Appendix, once more since certain of them seemed unavailable in the majority of texts. Complex variable theory is used mainly in respect of the elementary properties of the bilinear mapping of the plane. While the book has been written in what seemed the most logical order, and cross-references to analogs elsewhere are often made, it may also be read piecewise; however, Chapters 1-2, 4-6, and 11-12 form connected sequences. Problems have been given for each chapter. In most cases, these range from elementary exercises through straightforward generalizations to research suggestions. Little reference has been made to the use of functional analysis in connection with the boundary problems discussed here. In part, this is justified by the special character of our problems, and by the aim of obtaining the results in the most expeditious and simple manner. Apart from this, it may be questioned whether the suggestive value of the theory of boundary problems for functional analysis has been exhausted, having in mind here the theory of the symmetrizable operator and that of the fractional-linear recurrence relation or “J-contractive” matrix function. An exclusive reliance on the theory of the self-adjoint linear operator would at present have a limiting effect on the theory of our problems. I am indebted to a number of colleagues for their critical comments. For comments on the material in lecture and manuscript form I must thank Dr. C. F. Schubert and Mr. C. E. Billigheirner. For their careful reading of the proofs, in whole or in part, my especial gratitude is due to Professor J. R. Vanstone, and to Professor B. Abrahamson.
...
Vlll
PREFACE
Finally, it is my particular pleasure to acknowledge the cooperation and patience of Academic Press, Inc., and to express my appreciation of the consideration given to this work by them and by Richard Bellman, the Editor of this series of monographs. F. V. ATKINSON Madison, Wisconsin October, 1963.
Contents
PREFACE
...........................
V
INTRODUCTION 0.1. 0.2. 0.3. 0.4. 0.5. 0.6. 0.7. 0.8.
Difference and Differential Equations . . . . . . . . . . . . . . . . . . . 1 The Invariance Property . . . . . . . . . . . . . . . . . . . . . . . . 4 6 The Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . All-Pass Transfer Functions . . . . . . . . . . . . . . . . . . . . . . 8 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The General Orthogonal Case . . . . . . . . . . . . . . . . . . . . . 13 The Three-Term Recurrence Formula . . . . . . . . . . . . . . . . . . 15 The 2-by-2 Symplectic Case . . . . . . . . . . . . . . . . . . . . . 21
1-Boundary
Functions
1.1. I .2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10.
Finite Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . The Boundary Problem . . . . . . . . . . . . . . . . . . . . . . . . Oscillation Properties . . . . . . . . . . . . . . . . . . . . . . . . . Eigenfunctions and Orthogonality . . . . . . . . . . . . . . . . . . . The Spectral Function . . . . . . . . . . . . . . . . . . . . . . . . . The Characteristic Function . . . . . . . . . . . . . . . . . . . . . . The First Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . The Second Inverse Problem . . . . . . . . . . . . . . . . . . . . . . Moment Characterization of the Spectral Function . . . . . . . . . . . . Solution of a Moment Problem . . . . . . . . . . . . . . . . . . . . .
2-The 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.1. 2.8.
Problems for Rational 25 21 29 31 35 31 39 42
46 51
Infinite Discrete Case
A Limiting Procedure . . . . . . . . . . . . . Convergence of the Fundamental Solution . . . Convergence of the Spectral Function . . . . . Convergence of the Characteristic Function . . Eigenvalues and Orthogonality . . . . . . . . . Orthogonality and Expansion Theorem . . . . A Continuous Spectrum . . . . . . . . . . . . Moment and Interpolation Problem . . . . . .
ix
............ . . . . . . . . . . . . . ............. . . . . . . . . . . . . . ............ ............. ............ .............
55 57
60
62 63 61
10 71
CONTENTS
X
2.9. 2.10. 2.11.
A Mixed Boundary Problem A Mixed Expansion Problem Further Boundary Problems
3-Discrete 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8.
...................... ...................... ......................
74 76 81
Linear Problems
Problems Linear in the Parameter . . . . . . . . . . . . . . . . . . . . Reduction to Canonical Form . . . . . . . . . . . . . . . . . . . . . The Real Axis Case . . . . . . . . . . . . . . . . . . . . . . . . . . The Unit Circle Case . . . . . . . . . . . . . . . . . . . . . . . . . The Real 2-by-2 Case . . . . . . . . . . . . . . . . . . . . . . . . . The 2-by-2 Unit Circle Case . . . . . . . . . . . . . . . . . . . . . . The Boundary Problem on the Real Axis . . . . . . . . . . . . . . . The Boundary Problem on the Unit Circle . . . . . . . . . . . . . . .
. .
83 85 87 89 90 92 94 96
L F i n i t e Orthogonal Polynomials 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9.
The Recurrence Relation . . . . . . . . . . . . . . . . . . . . . . . . Lagrange-Type Identities . . . . . . . . . . . . . . . . . . . . . . . . Oscillatory Properties . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral and Characteristic Functions . . . . . . . . . . . . . . . . The First Inverse Spectral Problem . . . . . . . . . . . . . . . . . The Second Inverse Spectral Problem . . . . . . . . . . . . . . . . Spectral Functions in General . . . . . . . . . . . . . . . . . . . . . . Some Continuous Spectral Functions . . . . . . . . . . . . . . . .
. .
97 98 100 104 106 107 111 114 117
. . . . . . . . . . . . . . . . . . . . . ....... . . . . . . . ....... . . . . . . . ....... ....... . . . . . . .
119 120 123 125 129 130 132 134 136 138
. . . . . .
5-Orthogonal Polynomials The Infinite Case 5.1.
5.2.
5.3. 5.4.
5.5.
5.6. 5.7. 5.8. 5.9. 5.10.
Limiting Boundary Problems . . . . . . Spectral Functions . . . . . . . . . . Orthogonality and Expansion Theorem . Nesting Circle Analysis . . . . . . . . Limiting Spectral Functions . . . . . . Solutions of Summable Square . . . . . Eigenvalues in the Limit-Circle Case . . Limit.Circ1e. Limit-Point Tests . . . . . Moment Problem . . . . . . . . . . . TheDualExpansion Theorem . . . . .
6-Matrix 6.1. 6.2. 6.3.
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
Methods for Polynomials
Orthogonal Polynomials as Jacobi Determinants . . Expansion Theorems. Periodic Boundary Conditions . Another Method for Separation Theorems . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
142 144 145
6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10. 6.11.
CONTENTS
xi
The Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . A Reactance Theorem . . . . . . . . . . . . . . . . . . . . . . . . . Polynomials with Matrix Coefficients . . . . . . . . . . . . . . . . . Oscillatory Properties . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polynomials in Several Variables . . . . . . . . . . . . . . . . . . . . The Multi-Parameter Oscillation Theorem . . . . . . . . . . . . . . . Multi-Dimensional Orthogonality . . . . . . . . . . . . . . . . . . . .
148
.
150
150 152 157 160
.
162 169
....... . . . . . . . . . . . . . . . . . . . . ....... . . . . . . . ........ . . . .......
170 172 173 178 182 184 188 190 196 199
7-Polynomials Orthogonal on the Unit Circle 7.1. 7.2. 7.3. 7.4. 7.5. 1.6. 7.7. 7.8. 7.9. 7.10.
The Recurrence Relation . . . . . . . . . . . . . . . . . The Boundary Problem . . . . . . . . . . . . . . . . . Orthogonality . . . . . . . . . . . . . . . . . . . . The Recurrence Formulas Deduced from the Orthogonality Uniqueness of the Spectral Function . . . . . . . . . . . . The Characteristic Function . . . . . . . . . . . . . . . A Further Orthogonality Result . . . . . . . . . . . . . Asymptotic Behavior . . . . . . . . . . . . . . . . . . Polynomials Orthogonalona Real Segment . . . . . . . Continuous and Discrete Analogs . . . . . . . . . . . . .
8-Sturm-Liouville 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.1. 8.8. 8.9. 8.10. 8.11. 8.12. 8.13.
Theory
The Differential Equation . . . . . . . . . . . . . . . . . . . . . . . Existence. Uniqueness. and Bounds for Solutions . . . . . . . . . . . . . The Boundary Problem . . . . . . . . . . . . . . . . . . . . . . . . Oscillatory Properties ........................ An Interpolatory Property . . . . . . . . . . . . . . . . . . . . . . . The Eigenfunction Expansion . . . . . . . . . . . . . . . . . . . . . Second-Order Equation with Discontinuities .............. The Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . Convergence of the Eigenfunction Expansion . . . . . . . . . . . . . . . Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . Explicit Expansion Theorem . . . . . . . . . . . . . . . . . . . . . . Expansions over a Half-Axis . . . . . . . . . . . . . . . . . . . . . . Nesting Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . .
202 205 207 209 217 222 226 229 232 238 240 243 247
9-The General First-Order Differential System 9.1. 9.2. 9.3. 9.4.
Formalities . . . . . . . . . . . . . . . . The Boundary Problem . . . . . . . . . . . Eigenfunctions and Orthogonality . . . . . The Inhomogeneous Problem . . . . . . . . .
............. ............. . . . . . . . . . . . . . . .............
252 255 258 262
xii 9.5. 9.6. 9.7. 9.8. 9.9. 9.10. 9.11. 9.12.
CONTENTS
The Characteristic Function . . . . . . . . . . . . . . . . . . . . . . The Eigenfunction Expansion . . . . . . . . . . . . . . . . . . . . . Convergence of the Eigenfunction Expansion . . . . . . . . . . . . . Nesting Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansion of the Basic Interval . . . . . . . . . . . . . . . . . . . . . Limit-Circle Theory . . . . . . . . . . . . . . . . . . . . . . . . . . Solutions of Integrable Square . . . . . . . . . . . . . . . . . . . . . The Limiting Process a -P - m. b + m . . . . . . . . . . . . .
+
10-Matrix 10.1.
10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9.
..
268 273 280
..
289 292 293 298
284
Oscillation Theory
Introduction . . . . . . . . . . . . . . The Matrix Sturm-Liouville Equation . . . Separation Theorem for Conjugate Points . Estimates of Oscillation . . . . . . . . . Boundary Problems with a Parameter .. A Fourth-Order Scalar Equation . . . . . The First-Order Equation . . . . . . . . Conjugate Point Problems . . . . . . . . First-Order Equation with Parameter . . .
.
. . . .
. . . .
. . . .
. . . .
............. ............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..........
...............
300 303 308 312 317 323 328 332 336
11.-From Differential to Integral Equations
The Sturm-Liouville Case . . . . . . . . . . . . . . . Uniqueness and Existence of Solutions . . . . . . . . . . Wronskian Identities . . . . . . . . . . . . . . . . . Variation of Parameters . . . . . . . . . . . . . . . . Analytic Dependence on a Parameter . . . . . . . . . . Eigenvalues and Orthogonality . . . . . . . . . . . . . Remarks on the Expansion Theorem . . . . . . . . . . The Generalized First-Order Matrix Differential Equation . A Special Case . . . . . . . . . . . . . . . . . . . . 11.10. The Boundary Problem . . . . . . . . . . . . . . . .
11.1.
11.2. 11.3. 11.4. 11.5. 11.6. 11.7. 1 I .8. 11.9.
. . . . . . . . . .
. . . . . . . . . .
...... . . . . . . . . . . . . . . . . . . . . . . . . ...... ...... . . . . . . . . . . . . . . . . . .
339 341 348 350 355 356 358 359 363 364
12-Asymptotic Theory of Some Integral Equations 12.1. 12.2. 12.3. 12.4. 12.5.
Asymptotically Trigonometric Behavior . . . . . . . . . . . . . . . . The S.Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Non-Self-Adjoint Problem . . . . . . . . . . . . . . . . . . . . . . The Sturm-Liouville Problem . . . . . . . . . . . . . . . . . . . . . . Asymptotic Properties for the Generalization ofy" [ka g(x)]y = 0 . .
+ +
. .
366 371 375 381 384
...
CONTENTS
12.6. 12.7. 12.8. 12.9. 12.10.
Xlll
Solutions of Integrable Square . . . . . . . . . Analytic Aspects of Asymptotic Theory . . . . Approximations over a Finite Interval . . . . . Approximation to the Eigenfunctions . . . . . Completeness of the Eigenfunctions . . . . . .
. . . . .
........... 391 . . . . . . . . . . . . 393 . . . . . . . . . . . . 398 . . . . . . . . . . . . 408 . . . . . . . . . . . . 411
Appendix I. Some Compactness Principles for Stieltjes Integrals 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9.
Functions of Bounded Variation . . . . . . The Riemann-Stieltjes Integral . . . . . . . . A Convergence Theorem . . . . . . . . . . . The Helly-Bray Theorem . . . . . . . . . . Infinite Interval and Bounded Integrand . . . Infinite Interval with Polynomial Integrand . . A Periodic Case . . . . . . . . . . . . . . The Matrix Extension . . . . . . . . . . . . The Multi-Dimensional Case . . . . . . . . .
. . . . . . . . . . . .\ . ............. ............. ............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. ............. ............. ,
416 418 423 425 426 428 430 431 434
.
Appendix I1 Functions of Negative Imaginary Type 11.1. 11.2. 11.3.
Introduction . . . . . . . . . . . . . . . . The Rational Case . . . . . . . . . . . . . Separation Property in the Meromorphic Case
............. 436 ............. 437 . . . . . . . . . . . . . . 439
.
Appendix I11 Orthogonality of Vectors III.1.
III.2.
The Finite-Dimensional Case . The Infinite-Dimensional Case
..................... .....................
441 442
Appendix IV. Some Stability Results for Linear Systems Iv.1. Iv.2. IV.3. Iv.4. Iv.5.
A Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case of a Differential Equation . . . . . . . . . . . . . . . . . A Second-Order Differential Equation . . . . . . . . . . . . . . . . The Mixed or Continuous-Discrete Case . . . . . . . . . . . . . . . The Extended Gronwall Lemma . . . . . . . . . . . . . . . . . . . .
447
. . 449 . . 450 . . 452 455
xiv
CONTENTS
Appendix V. Eigenvalues of Varying Matrices
................ . . . . . . . . . . ... . . ............... ... . .. . . ..... . .. .......... ... . . . . . . . .. . .. . ..
Variational Expressions for Eigenvalues . Continuity and Monotonicity of Eigenvalues A Further Monotonicity Criterion . . . . Varying Unitary Matrices . . . . . . . . Continuation of the Eigenvalues . . . . . V.6. Monotonicity of the Unit Circle . . . . .
V. 1. V.2. V.3. V.4. V.5.
,
457 459 461 464 465 468
Appendix VI. Perturbation of Bases in Hilbert Space VI.1. VI.2. VI.3.
The Basic Result . . . . . . . Continuous Variation of a Basis. Another Result . . . . . . . .
NOTATION AND TERMINOLOGY
........................
LIST OF BOOKS AND MONOGRAPHS
NOTES:
. .. . .. .
471 473 475
,
476
.................
478
Section 0.1, Section 0.2, Section 0.4, Section 0.7, Section 0.8, Section 1.5, Section 1.6, Sections 1.7-8, Section 1.10, Section 2.2, Section 2.3, Section 2.5, Section 2.7, Section 2.10, Section 3.1, Section 3.2, Section 3.3, Section 3.5, Chapter 4, Section 4.2, Section 4.3, Section 4.4, Section 4.5, Section 4.7, Chapter 4, Problems, Section 5.1, Section 5.2, Section 5.4, Section 5.7,Section 5.8, Section 5.9, Section 5.10, Section 6.1, Section 6.4, Section 6.5, Section 6.6, Sections 6.9-10, Section 7.1, Section 7.5, Section 7.6, Section 7.7, Sections 7.8-9, Section 7.10, Section 8.1, Section 8.3, Section 8.4, Section 8.5, Section 8.6, Section 8.7, Section 8.10, Section 8.13, Section 9.1, Section 9.4, Section 9.11, Section 10.1, Section 10.2, Section 10.3, Section 10.4, Section 10.6, Sections 10.8-9, Section 11.1, Section 11.2, Section 11.8, Chapter 11, Problems, Section 12.1, Section 12.3, Section 12.4, Section 12.5, Section 12.6, Section 12.7, Section 12.10, Appendix I, Appendix 11, Appendix'IV, Appendix V, and Appendix VI . 481
PROBLEMS.. INDEX
.. ...... .. .. ........ . . , .. .. . ... . . . .. .. .. . . . .. .. . . . . . . ... .. ..
. ... .
.
..........................
...................................
536 565
Introduction
0.1. Difference and Differential Equations
Our problems in the sequel belong to the field of linear analysis, in that the unknown functions or their derivatives appear only to the first power in the governing equations. They are also linear in a second sense, or perhaps sequential, in that the “independent variable” ranges over some part of the real line, perhaps over a discrete series of values. When a physical interpretation is available, it will be in terms of entities ranged along a line, each of which interacts only with those in its immediate vicinity. Alike on the physical and mathematical sides, it is convenient to devise separate treatments for the purely discrete case and for the purely continuous case. I n the former case the situation is that there is a discrete sequence of entities, each of which either gives an input to its successor, or perhaps interacts with both its neighbors; this is immediately describable by means of a differencc or recurrence relation, often soluble by algebraic processes. I n the case when the physical entities are not separate, but form some continuous distribution, limiting processes of a more or less satisfactory character lead us to differential equations. I n both cases, the wealth of the mathematical theory amply justifies a concentration on these two cases separately. When the topics of difference and differential equations are developed separately, many analogies are observed between the results. These analogies form our main concern here, in so far as they apply to boundary problems. It should be emphasized that these analogies result from specializing a single situation in two directions. While such specialization may well be necessary in order to accumulate usetul results, neither the difference equation nor the differential equation provides a fully adequate framework for the topic of boundary problems. I n the case of differential equations, this inadequacy is to some extent apparent in the need to consider “weak solutions,” or to use the machinery of “distributions.” 1
2
0. INTRODUCTION
Our treatment in what follows will be a compromise between the total separation of difference and differential equations, on the one hand, and their unification within a more general but necessarily more abstruse theory, on the other. T h e simplifying features of difference or recurrence relations are not to be wasted; on the other hand, we shall give occasional attention to what may be termed the mixed continuousdiscrete case, which combines features of both difference and differential equations. Though we do not essay here a general treatment of such mixed continuous-discrete cases and have referred to such a treatment as necessarily abstruse, the underlying physical idea is quite simple. Returning to the notion of interacting entities ranged continuously or discretely, or both, along a line, the physical or other hypotheses concerning laws of motions will tell us that the change in some quantity from one point of the system to another point depends on the state of the system between the two points. This situation is naturally expressed by an integral equation, involving a Stieltjes integral in the discrete or the mixed case. T h e formation of a differential equation by a limiting process is a step which may be dispensed with, so far as the general theory is concerned. T h e close relation between discrete and continuous cases, and the possibility of subsuming the two within a larger theory, will be evident from the physical systems which realize certain of our boundary problems. An example of great suggestive value is given by the vibrating string, stretched between two fixed points, loaded in some manner, and executing small harmonic vibrations. If the string itself be weightless and bears a discrete sequence of particles, perhaps a finite number, then the amplitudes of oscillation of three consecutive particles are connected by a recurrence relation. At the other extreme, we suppose that the string itself has a continuously varying density, and bears no particles, deriving in this case a second-order differential equation. It is plausible that this heavy string might be approximated, in dynamic behavior, by a finite number of suitably distributed particles; in effect we approximate here to a differential by a difference equation. For the general case, it is physically intelligible that the string should both have a density itself and bear particles in addition; this calls for an integral or integro-differential equation, including differential and difference equations as special cases. Similar remarks apply to an electrical interpretation. If as the continuous problem we take the harmonic propagation of a disturbance in a transmission line formed of two parallel conductors, the discrete analog will be that of propagation through a series arrangement of fourterminal networks. As regards approximation between the two cases,
0.1.
DIFFERENCE AND DIFFERENTIAL EQUATIONS
3
we might wish to simulate a transmission line by a dummy line formed of a finite number of circuits, or by making the number of such circuits become larger, we might derive the transmission line equations. For the general case, we clearly have a problem in the same area if we interpose networks at points along a long line. A further example of propagation by continuous or discrete steps is given by propagation through a stratified medium, where the properties of the medium may vary continuously, or abruptly, or both. For mathematical purposes, the difference equation often appears as an aid to the study of a differential equation. An outstanding classical example is provided by the initial-value problem. T o have a definite case in front of us, let us take the second-order differential equation. y"
+ hp(x)y = 0,
0 Q x Q 1,
(0.1.1)
where p ( x ) is, say, continuous; the initial-value problem calls for a solution such that y(0) = a, y'(0) = b. (0.1.2) The approximation
Y"(4
-
h-Yy(x
+ h) - 2Y(X)+ y(x - h))
(0.1.3
suggests approximating to (0.1. l ) by
+ h) - 2yt(x) + yt(x - h)} + h p ( ~yt(x) ) = 0.
h-'{yt(~
(0.1.4)
For convenience we suppose h = m-l for some integer m, and write yn = yt(nh); it then follows from (0.1.4) that y n f l - 2y,, +Y,,-~
+ hh2p(nh)y, = 0 ,
n = 1, ..., m - 1.
(0.1.5)
For the initial conditions we approximate to (0.1.2) by Yo
= a,
(Yl -Yo)lh
= b.
(0.1.6)
Then yz,y3, ... are determined recursively from (O.lS), and we have replaced the initial-value problem (0.1.1-2) by the initial-value problem (0.1.5-6),the latter being immediately soluble. The conjecture to be tested is then that yn = yt(nh) approximates to y(nh), where y ( x ) is the solution of (0.1.1-2). This, or similar approaches, provides the basis of both a computational solution of the initial-value problem (0.1.1-2), and also of existence theorems for the existence of a solution of (0.1.1-2). The same ideas may be applied to boundary problems. I n the simplest
4
0.
INTRODUCTION
of boundary problems for (0.1. l), we ask for A-values, the eigenvalues, such that (0.1.1) has a solution such that Y(0) = Y U )
= 0,
(0.1.7)
without, however, y ( x ) vanishing identically, and so subject to y’(0) # 0. In the parallel problem for (0.1.5), we ask for A-values such that (0.1.5) has a solution such that yo = y m = 0,
(0.1.8)
and again not vanishing throughout, and so such that y1 # 0. It is natural to conjecture that for small values of the tabular interval h, that is to say, for large m,at least the smaller eigenvalues of (0.1.5), (0.1.8) approximate to those of (O.l.l), (0.1.7), and that the eigenfunctions, the corresponding sequences satisfying (0.1.5), (0.1.8) can be used as a basis for approximation to the eigenfunctions of (0.1. l), (0.1.7). This latter argument has been used from time to time as a means of proving the eigenfunction expansion for (0.1.l), (0.1.7). Since the eigenfunction expansion for (0.1.5), (0.1.8) is purely elementary, being in effect the expression of an arbitrary vector as a linear combination of the eigenvectors of a symmetric matrix (see Chapters 4 and 6), this method provides perhaps the most elementary proof of the SturmLiouville eigenfunction expansion. It can also serve as a foundation for the eigenfunction expansion when (0.1.1) is generalized to allow certain discontinuities. In the reverse direction, it is possible to represent (0.1.5) in terms of a differential equation. We use this in Chapter 8 to give a unified treatment. Our attitude will be that boundary problems for difference and for differential equations are equally deserving of study, and that each can be of suggestive value for the other.
0.2, The Invariance Property Our boundary problems will have the properties associated with “self-adjoint” problems, in that eigenvalues will be real, and there will be an orthogonality of eigenfunctions. Without laying much stress on the concept of self-adjointness, we adopt as the key property that the differential or difference equation should admit, for real parametervalues, an integral of a certain form. I n the case of (0.1.1) this will be the well-known fact that the Wronskian of two solutions is constant.
0.2. THE
INVARIANCE PROPERTY
5
In general, there is to be a quadratic form in the solution which is constant, in the independent variable. If we postpone for the moment a detailed exposition of this requirement, it may be remarked that such a property is to be expected on heuristic grounds in connection with vibrating physical systems which are nondissipative ; for the vibrating string, for example, the particles are not subject to any friction, whereas for our cascade of circuits only LC-circuits are to be used. Supposing that the system is oscillating harmonically, we concentrate attention on the energy, potential and kinetic, located in some segment of the system. Since the time average of the energy in this segment is constant, the time-average of the energy flow into the segment at one end must equal the average energy flow out at the other end. Thus the average energy flow is the same at all points along the system, and this is the invariant quadratic form in question. We outline the latter argument in the case of (0.1.1). T h e property is that, if h is any real quantity, then a solution y ( x ) of (0.1.1) satisfies y’jj
-95
=
const,
(0.2.1)
where y(x) is the complex-conjugate solution. That this is so may, of course, be verified by differentiation of (0.2.1) and use of (0.1.1). However, we may also derive it, for positive A, by considerations of energy. Starting from the differential equation of a vibrating string in the form (0.2.2) we seek solutions of the form u(x, t ) = y(x) exp (-i
dit ) ,
(0.2.3)
so that y(x) must satisfy (0.1.1). We now allow y to be complex, the physical disturbance being measured by the real part of u. This becomes a standard procedure, if we shift the metaphor to electromagnetic theory; we interpret E = (0, u, 0) as the electric vector of a linearly polarized wave moving in the x-direction in a stratified dielectric medium with parameters E = E(x), p = constant, with in (0.2.2)p(x) = c(x)p. T h e magnetic vector is then
H
=
(0,0,(ip dX)-’uZ),
(0.2.4)
where u, = &/ax = y’ exp (- i f i t ) , the complex Poynting vector being
i E x H = &+y’/(p
dX), 0,0).
(0.2.5)
6
0. INTRODUCTION
T h e real part of this, namely (0.2.6)
*(t.di)-l(zjy- iyy’, 0,O)
is conventionally interpreted as the mean energy-flow through unit area; in our present case it must be constant in x, since no energy is dissipated. In this way, the Wronskian property (0.2.1) becomes a case of Poynting’s theorem. In order to obtain a broader perspective, we put this latter result in matrix terms. To rewrite in such terms the original differential equation (0.1.1) we define the column matrix or vector =
so that (0.I . I ) yields z’ =
t:,) ( =
(;,j
Y’ --hPY
) = (+0 );
(0.2.7)
z.
(0.2.8)
T h e (*) is used to indicate the complex-conjugate transpose, so that z* is the row matrix (99’).Then (0.2.1) may be written compactly as where
z*Jz = const, ]=(O 1
-1)0 .
(0.2.9) (0.2.10)
Very similar identities to (0.2.1) are known in connection with selfadjoint differential operators of the nth order, associated with the topics of the Lagrange identity or of Green’s theorem. When translated into matrix terms, these will involve nth order matrices in place of (0.2.10). We consider some cases in Chapter 9. Such identities also occur in discrete cases such as (0.1.5) (see, for example, Section 4.2). In what follows we shall regard identities of the type of (0.2.9) as a means of classifying, and to some extent as a means of setting up boundary problems. We proceed to what is perhaps the simplest case of a boundary problem.
0.3. The Scalar Case In keeping with our program of setting u p boundary problems in association with invariance properties, we shall do this first for the case in which the invariance property merely says that a complex number has constant modulus. This will be the case of (0.2.9) in which all quantities
0.3.
7
THE SCALAR CASE
are scalars, J being the 1-by-1 matrix unity. There will again be discrete, continuous, and mixed cases. Of these the continuous case is well known and is given by the differential equation y' = ixy
[(')
(0.3.1)
= d/dx],
where X is a parameter a n d y a scalar dependent variable. It is immediately verifiable that if h is real, there holds the invariance property jjy = const.
(0.3.2)
T h e boundary problem associated with (0.3.1) is also well known. We choose a real 01, 0 01 < 27r, and require that (0.3.1) have a solution, not identically zero, such that
0, I y(1) I > I y(0) 1 when I m h < 0, which proves the result. T h e discrete analog of this situation will deal with a sequence y o ,yl, ... connected by a recurrence relation
which is to have the property that
I yn+1 I
(0.3.5)
= I Yn I
if X is real. This cannot be achieved by making &(A) a linear function of A, and the simplest function with this property will have the fractional linear or bilinear form &(A) = (1
+ iACn)/(l- Atn),
(0.3.6)
where c,, E, are complex conjugates, and we have arranged that +%(O) = 1. For the corresponding boundary problem there will first be the finite discrete case, in which we have a finite recurrence formula Y,,+~ = yn(l
+ iAcn)/(l- i&),
n
= 0,
..., m
and again, in a similar manner to (0.3.3), require that
- 1,
(0.3.7)
8
0.
INTRODUCTION
T h e proof that the eigenvalues are all real proceeds much as before, assuming that the cn all have positive real part: we take this up in Chapter 1. There will also be a limiting boundary problem of discrete type with m = 00, which we consider in Chapter 2, and mixed continuousdiscrete problems. Associated with the boundary problem (0.3.1), (0.3.3) there will be an orthogonality for the eigenfunctions and an expansion theorem. I n the special case a: = 0, these become the orthogonality and expansion theorem associated with complex Fourier series; the case of general a: will be similar. For the discrete recurrence formula (0.3.7) with the same boundary conditions, the eigenfunctions and orthogonality relations will concern certain rational functions, of which we give details in Chapter 1. There will again be a possibility of approximating to the continuous differential equation (0.3.1) by means of discrete relations (0.3.7), in other words of expressing exp (ih) as the product of factors (0.3.7). Although it is fairly obvious that this can be done, the problem of making such an approximation in a best possible manner has some practical interest. 0.4. All-Pass Transfer Functions
T o illustrate the boundary problem just discussed, we borrow notions from the topic of servomechanisms. Writing (0.3.7) in the form Yn+l
- i%Yn+l
= Yn
+ ihcnyn
9
(0.4.1)
we are led to consider the differential recurrence relations (0.4.2)
which we may imagine as a sequence of devices of which the nth feeds into the (n 1)th. We suppose that there is applied a sinusoidal driving force uo = yo exp ( - A t ) , where X is real and y o is a non-zero constant. We find, for example, that
+
and so
+ t0zi,= (1 + ihc,) yo exp ( - a t ) , u1 = (1 + ihc,) (1 - iht0)-l y o exp ( - A t ) , u1
(0.4.3) (0.4.4)
where we have omitted as exponentially damped a term in exp ( - t / F o ) . Proceeding in this way we find that urn = Y m ( 4 Yo exp (-W
(0.4.5)
0.4.
9
ALL-PASS TRANSFER FUNCTIONS
where (0.4.6)
T h e function ynL(A)just defined expresses the ratio of the output unb to the input uo = yo exp (-iAt), and is the “transfer function” of the system. Since
IYm(4 I
=
(0.4.7)
1
for all real A, as may be seen from the form (0.4.6), the driving function yo exp (-iAt) passes through the system with no change in absolute value; the term “all-pass” is used for such a transfer function. T h e effect of the system is thus to apply a phase shift (0.4.8)
(0.4.9)
Our boundary problem (0.3.8) requires the phase shift to have a specific value. More precisely, determining ym(A)from (0.4.9) as a continuous function for real A, fixed by qn,(0)= 0, the eigenvalues are the roots of ym(h)= a (mod 27~).
(0.4.10)
We may interpret these eigenvalues as the singularities of a feedback problem with phase shift a ; the case a = T is commonly studied in applications. I n place of (0.4.2) we set the system
+
u*+~
= u,
- c,&,
n = 1,
..., m
-
1,
(0.4.1 1)
together with the feedback equation u1
+ foCl = (1 - co d / d t )(uo + e-”Lu,).
(0.4.12)
Again we consider the result of applying a driving function uo = y o exp (-iht), and select the solution, if any, of (0.4.12) of the form const. exp (-iht). Putting zi, = -iAu, to find such a solution, we may obtain from (0.4.11-12) (0.4.13) (0.4.14)
10
0.
INTRODUCTION,
Here y,(h) is as given by (0.4.6). T h e effect of the system is now to multiply the driving function yo exp (-iht) by the “closed-loop transfer function” Ym(h)/[l - e-”?n(4l. (0.4.15) This function has poles at the eigenvalues, which are accordingly the values of h for which the system exhibits a singular response to the driving force exp (-;At). Functions of a very similar type to (0.4.15) will be termed later “characteristic functions,” and will play a basic role in all our boundary problems. They are closely related to two important concepts, that of the spectral function and that of the Green’s function. We may modify this feedback problem so as to obtain a more exact analog of the characteristic and the Green’s function. Recalling that the latter is, in essence, the ratio of the response of the system at one point to a disturbance applied at the same or another point, roughly speaking, we set up the problem
+
u,,+~ E,U,+,
= un - cnUn,
n = 1, ..., m - 1,
+ exp (-&I}, (I - co d / d t ) + + exp (-iht)}.
u, = exp (ia) {uo -
u1
+ F~U,=
{uo
(0.4.16) (0.4.17) (0.4.18)
Here we have introduced an inhomogeneous term exp (-iht), distributed half on either side of the “point of application” uo, treating in (0.4.17) the boundary condition as a member of the family of recurrence relations. If we select again the solution, if any, of the form const. exp (-iht), and so put 14, = on exp (-;At), the above equations yield v,+, w,
= (1
= e“(w0
+ ihc,) (1 - i h ~ , ) - ’ w , , - +),
wl = (1
n = 1, ..., m - 1,
+ ihco) (1 - iAt0)-1 (wo + 4)
whence, with the notation (0.4.6),
T h e solution of (0.4.16-18) is, therefore, in part,
T h e coefficient of exp (-;At) on the right may be considered, moving to network terminology, a driving-point admittance, as the ratio of response at the location uo to the force exp (-;At) administered there.
0.5. INVERSE PROBLEMS
11
Modifying this function by a constant factor, we shall consider in Chapter 1 the function f m . m
=
(W1+
mt71(h))/{ei0! - ym(X)>r
(0.4.20)
terming this the characteristic function, in analogy with a partial usage for differential equations. It has the important properties that it is real when h is real, since then I yl(h) I = 1, with poles at the eigenvalues, and has negative imaginary part when I m h > 0. It is also related, by way of the Stieltjes transform, to the spectral function, which we define later. In a similar way we may find u1 , u2 , ... from (0.4.16-18), extending (0.4.19) to u, = exp (-Lit) eiyn((x)/{eia - ym(h)},
(0.4.21)
the coefficient of exp (-iht) being a transfer admittance. We may next repeat the whole process with the “driving force” applied to the location ul, according to
+,flu2 (I - c1 d/dt){ul + 9 exp (-iht)}, (1 + E, dfdt) {ul - 8 exp (-&)I u, - cogo, u2
=
=
the remainder of the recurrence relations and the boundary condition being unaltered. This will yield another driving-point admittance and transfer admittances. If generally g,.,(h) is the ratio of the function u, to a disturbance exp (-iht) applied in the case of ur in the above manner, the matrix g&i) may be considered as a Green’s function.
0.5. Inverse Problems In widely separated contexts, inverse problems present themselves. Here there is a boundary problem in which the differential or difference equation is unknown. The information given us may consist of the eigenvalues for a known boundary condition, or perhaps several sets of eigenvalues for several boundary conditions. A related problem is that in which the asymptotic behavior of the solutions is given, the differential or difference equation to be found. The problem may appear as one of design or synthesis of an apparatus with a view to some prescribed performance, or as one of diagnosis, as when the “potential” in the Schrodinger equation is to be found from scattering measurements. To give an illustration at a purely algebraic level, we might try to
12
0.
INTRODUCTION
find the constants c, implicit in (0.4.6),given the singularities of the closed-loop transfer function (0.4.15); as a matter of fact, we should need two sets of singularities corresponding to two values of a, these singularities having the separation property. We deal with this in Section 1.8. A closely analogous problem is that of determining a set of orthogonal polynomials, and so a recurrence relation, given the zeros of two consecutive polynomials of the set. From the theoretical point of view, when such inverse problems have been solved, sometimes no mean task, we seen the theory in a finished form. Necessary and sufficient conditions are then known in order that the differential or difference equation should lead to eigenvalues of some specified character, or should have solutions with some specified asymptotic behavior. Similar inverse considerations may be applied to intermediate stages in the theory of a boundary problem. For example, in the simplest case of the finite discrete scalar problem (0.3.7-8), the recurrence relation is essentially contained in the transfer function ym(h) given by (0.4.6), the eigenvalues being the roots of ym(h)= exp (2.). From the functiontheoretic point of view, y,,(h) is a rational function which maps the real A-axis into the unit circle, and the upper and lower half-planes into the inside and outside of the unit circle, respectively, the c, being assumed to have positive real part. I t may be shown that the form (0.4.6) gives the most general rational function with these properties, subject to Y r m = 1. This intrinsic characterization of ym(h) as a rational function with specified mapping properties gives our boundary problem, within its limits, an air of finality. It also suggests characterizing some related classes of boundary problems in this way. For example, we might ask for the most general entire function mapping the upper and lower half-planes into the inside and outside of the unit circle. T h e answer is given by a complex exponential, which may be interpreted as the transfer function for the differential equation (0.3.1). Combining these two cases, we might next ask for the most general meromorphic function with these mapping properties, being led to a combination of (0.3.1), (0.3.7) into the mixed continuous-discrete case. Still further generality is obtained if we merely consider functions mapping the upper halfplane into the interior of the unit circle. I n the problems just discussed, there is little difficulty in principle in proceeding from, in the discrete case, the transfer function (0.4.6) to the recurrence relation (0.3.7). T h e corresponding higher-dimensional problem, of the factorization of contractive matrix functions is much more substantial.
0.6.
THE GENERAL ORTHOGONAL CASE
13
0.6. The General Orthogonal Case T h e further course of the theory of boundary problems might be baldly summarized as the extension of these ideas into vector and matrix terms. I n each case we may start with a differential or a difference equation, leaving some quadratic form invariant, and involving a parameter. Here we outline the case in which it is to be the length of the vector which is invariant. We start with the continuous case, and give a direct extension of the differential equation (0.3.1). This is given by the first-order system y’ = ihHy,
0
< x < 1,
(0.6.1)
where now y = y(x) is a k-vector, written as a column matrix with entries yl(x), ...,yk(x), and H = H ( x ) is a k-by-k matrix of functions of x, which for convenience we may suppose continuous; all quantities may be complex. T h e desired invariance property is that y ( x ) should be of constant length, in the ordinary sense, so that
2 ~y,(x) k
12
=
r=1
const,
o < x < I.
(0.6.2)
Denoting by y* = y * ( x ) the complex conjugate transpose of y ( x ) , that is to say, a row matrix with entries y l ( x ) , ..., yk(x), we may write (0.6.2) more compactly as y*y = const, 0 < x < 1. (0.6.3) This forms an extension of (0.3.2), or again a particular case of (0.2.9). We require that the matrix H should be such that (0.6.3) is true for any solution of (0.6.1) if X is real. This is ensured if H ( x ) is Hermitean, or equal to its complex conjugate transpose, H=H*,
(0.6.4)
O<x 0,
0 <x
< 1,
(0.6.10)
in the sense that H ( x) is positive-definite, it follows from (0.6.8) that Y*Y is a decreasing function of x if I m h > 0, and increasing if Im X < 0. Thus Y * ( l )Y ( l ) < E, Im h > 0, (0.6.11) Y * ( l )Y ( l ) > E,
Im h
< 0.
(0.6.12)
We may interpret the matrix loci in (0.6.9), (0.6.11), and (0.6.12) as the (matrix) unit circle, and its interior and exterior, respectively, and again assert that the transfer function Y(l) maps the upper and lower half-planes in the X-plane into the interior and exterior of the (matrix) unit circle. For a boundary problem, similar to (0.3.3), we take some fixed unitary matrix N , say, with N*N = El and ask for what X-values (0.6.1) has a nontrivial solution such that Y(1) = W ( 0 ) ;
(0.6.13)
we could, of course, put N into exponential form, as was done in (0.3.3), which might be preferable for the study of separation theorems. Here we confine ourselves to observing that the eigenvalues are necessarily real; by (0.6.13) and the fact that N is unitary, y(1) and y(0) have the same length, and sincey(1) = Y(l)y(O), this is impossible by (0.6.11-12) if X is complex. For a discrete boundary problem associated with the invariance property (0.6.3), we seek elementary factors, matrix functions of A,
0.7.
15
THE THREE-TERM RECURRENCE FORMULA
which are unitary when h is real, and lie, so to speak, inside the unit circle when h is in the upper half-plane. Such factors are once more, as in (0.3.6-7), given by bilinear expressions. Denoting by yo, yl, ... a discrete sequence of K-vectors, we are led to set up the recurrence relations n = 0,1, ... , (0.6.14) yn+l= ( E iAC,) ( E - iACn*)-lyn,
+
+
where the C, are “normal,” C,*C, = CnCn*, and C, C,* > 0, or have positive “real part.” For the boundary problem we consider again whether there is a solution such that ym = N y , , where N is a fixed unitary matrix. There are a number of variants on the two boundary problems just formulated, of which a bare mention must suffice: (i) the boundary problem (0.6.1), (0.6.13) may be studied, in the limit, over the interval 0 x < a, or - 03 < x < a,yielding “limitpoint” and “limit-circle” cases ; (ii) the recurrence relation (0.6.14) may be studied over 0 n < m, or over < n < m, again with a discrimination between limit-point and limit-circle cases; (iii) the variation of y ( x ) may be determined in part by a differential equation, in part by a recurrence relation, according to a scheme of the form, for example,
- B r { P r s ( t )
- Pr-l,dt)}s
(0.7.16)
which again is close to (0.7.4). Again, more informative expressions for P(t) than (0.7.15) may be found in terms of the eigenvalues and eigenvectors of T , effectively in terms of certain orthogonal polynomials and their zeros. In the above we have confined ourselves to problems which are finite and discrete. Extensions that suggest themselves include in the first instance infinite discrete cases. T h e vibrating light string might bear an infinity of particles, with one or more limit-points, finite or otherwise, which is, of course, included in the general case in which the string also has weight. 0.8. The 2-by-2 Symplectic Case
We may use this as a general term to cover cases exhibiting an invariance of the form (0.2.9-lo), or again (0.7.3). T h e Sturm-Liouville
22
0. INTRODUCTION
case of (0.1.1) or (0.2.8) is included in that of a two-dimensional system of the form Ul’(4 f l z w
+
= {Jw44 4 ) u2(4, = - {hq2(x) T Z ( X ) ) u,(x),
+
1
o<x, 0, since otherwise the points i/cp would not lie in the upper half-plane. From (1.6.6) we see thatf,,,(A) takes complex conjugate values at complex conjugate A-values, so that Jrb,,(h)= - 1/(223 when h = - i/Fp. These latter points must be poles of yL(X), by (1.7.1). Thus
where c is a constant. T o show that c = 1, in accordance with (1.3.1), we verify that ym(0)= 1. In fact, it follows from (1.6.3) that fnl,.(0)= - Q cot 8 a, and substituting this in (1.7.1) we do indeed get yL(O) = 1, as asserted. It remains to consider whether if we set up the boundary problem (1.2.2-3) with the co , ..., c,-~ thus found, and find eigenvalues and normalization constants as in Section 1.4, we obtain those originally prescribed. We must arrive at the same characteristic function fm,,(h), since these are connected by the one-to-one relationship given in (1.6.1) or (1.7.1). Since the eigenvalues and normalization constants are uniquely fixed when the characteristic function is known, these must be the same as those prescribed. We pass to the exceptional indeterminate case.
Theorem 1.7.2. I n the notation of Theorem 1.7.1, let the h, be distinct real numbers of which exactly one is zero, and let the p r be positive. Then there is a one-parameter family of sets of values of c,, , ..., c , ~ - ~such that the boundary problem (1.2.2-3) with 01 = 0 has the A,, as its eigenvalues and the p,, as its normalization constants. In this case we define f?)l,a(h)by (1.6.2) or (1.6.6), leaving ym,, as an indeterminate. As previously, ynl(h)has m zeros and m poles, lying at
42
1.
BOUNDARY PROBLEMS FOR RATIONAL FUNCTIONS
complex conjugate pairs in the upper and lower half-planes, respectively. T h e only difference is that we prove that y,(O) = I by making A + 0; as A ---t 0 in (1.6.6) we have fm,=(A) ---t 03, and inserting this in (1.7.1) we see that y,,(h) ---t 1, so that ~ ~ ( =0 1.) Since Y , , , ~ may now have any real value, there is as stated a one-parameter set of.solutions to our problem.
1.8. The Second Inverse Problem We have mentioned that the boundary problem is not determined by a knowledge of its eigenvalues; the eigenvalues give roughly half the information necessary. It is plausible that the boundary problem might be fixed when we are given two sets of eigenvalues, for two given boundary conditions and one and the same, unknown, recurrence relation. We shall term this the second inverse spectral problem, the first being that in which we are t‘old the spectral function. We treat here the main case in which none of the eigenvalues is either infinite or zero. T h e oscillation theorems of Section 1.3 impose certain restrictions on the assumed eigenvalues. They must have the separation property, by Theorem 1.3.6. There is also a restriction on the eigenvalues nearest to zero.
Theorem 1.8.1. Let A,, p, be two sets of m real numbers, all 2m numbers being distinct, finite and non-zero. Let them have the separation property that between any two of the A, there lies at least one of the p, , and conversely. Let the closest to zero among the A,, pr be one of the A,, one the positive side (if any of the 2m numbers are positive), and one of the pr on the negative side (if any of the 2m numbers are negative). Let a, a‘ be given with 0 < a < a’ < 2n. Then there is a set of complex numbers co , ..., cmP1,unique except as to order, such that the boundary problem (1.2.2-3) has the A,, as its eigenvalues, and with a’ replacing a has the p, as its eigenvalues. We have that the equations ym(A)= exp (ia),y,,(h) = exp (id)have as their roots the A,, , p, , respectively. With the notation
n(1+ m-1
n 1 =
k=O
n(1 m-1
ihck),
17, =
-
k-0
;he,),
(1.8.1) (1.8.2)
we have then the identities exp (ia)II, = [l - exp (ia)]113 , If,- exp (id)L7, = [I - exp (id)] l7, ZI,
-
.
(1.8.3) (1.8.4)
1.8.
43
THE SECOND INVERSE PROBLEM
T o prove (1.8.3), we observe that the left-hand side is equal to (Yrn(X)
- ~ X P(i.1)
n, >
and is therefore a polynomial of degree m with the zeros A, . It is therefore a multiple of 17, , and the constant factor on the right of (1.8.3) is obtained by equating the constant terms on both sides. The proof of (1.8.4) is similar. Here 17,,Il,and a, 01' are all known, and so l7,and 17,are determined from (1.8.3-4) by solving the equations. Since yrn(X) =
nl/n,
9
the boundary problem is recovered. It requires greater trouble to verify that the solution so obtained yields a set of ck with positive real part. These ck are specified by the fact that the zeros of ym(h),that is to say, the zeros of 17, , are the points ilc,. What we have to prove is that the zeros of l7, lie in the upper half-plane. For this purpose we use the argument principle. We consider the variation in arg17, as h describes a large semicircle, center the origin and diameter on the real axis, with curved portion in the upper halfplane, and described positively. If, as we say, the zeros of l7, all lie in the upper half-plane, then the variation in arg l7, around this contour will be 2mr. So far as the curved portion of the semicircle is concerned, on which l7, is asymptotic to a constant multiple of Am, the variation in arg17, will approximate to mr, for large semicircles. Hence we have to prove that the variation in arg l7,as h describes the real axis positively amounts to an increase of mr. Solving (1.8.3-4) for l7,, we have sin &(a' - a) l7, = -exp ( & i d )sin 8.
.n3+ exp ( i i a )sin &a' n,, (1.8.5) *
In this equation we put X equal to the successive 4, pr to study the variation of arg17,. It is necessary to ascertain the signs of the n3, l7, at the zeros of the other. T o illustrate the latter we draw up a table; we assume the A,, , pr numbered in the following manner : A_,
< p-v < ... < A_, < p-, < 0 < h, < po < ... < p m - p , < (1.8.6)
This expresses our assumptions that the 4, pr separate one another, and that the least positive such number belongs to the A,. , the greatest negative one to the p, . The possibility is not excluded that all the A,, ,
1.
44
A
BOUNDARY PROBLEMS FOR RATIONAL FUNCTIONS
A_,
p-,
sgnn,
0
(-)l-,
sgnn,
(-)p
0
... A_,
p-1
0
... 0 + + ... - 0 +
A,
p,
...
pm-9-1
Am-,
0
-
...
(-)-
0
...
0
(-)m-P
+ o
In the central column we record the fact that 17, , n4are both positive, in fact, unity, when h = 0. I n tabulating the signs of n3,for example, we use the fact that it vanishe's at each of the A,, , changing sign at each of them since it has only simple zeros. At two successive p,, ,17,will have opposite signs, since between them must lie just one of the A,,. We repeat that in the above table, of the first and last columns only one is actually present. Suppose for definiteness that A_, , A, both occur, and consider the variation in arg Ill as X increases from A_, to A, . We use (1.8.5), noting that sin (a' - a), sin a, sin a' are all positive, since 0 < a < a' < 2n. Putting A = A_, in (1.8.5), so that 17, = 0, l7, < 0, it follows from (1.8.5) that we may take arg I7,(Ll) = Q a - n. Making now h increase to p-, , we get 17, > 0,U4= 0, so that arg I7,(p-,) = Q a' - n(mod 2n). We assert that in fact arg l7,(p-,) = $a' - n. T h e alternative possibilities are that as X goes from A_, to p-, , arg17,(h) might go from a - n to 4 a' n, or to Q a' - 3n, or to more distant values still. If this were so, arg 17,(X) would reach in between the value 4 a', or a' - 2n or some value congruent to these (mod 2n). This would mean that the left of (1.8.5) and the first term on the right would have argument congruent to & 01' (mod n), the second term on the right having argument a (mod n). Since 0 < 01 < a' < 2n, this is not possible unless U , = 0, and this too is impossible between A_, and p-,. We conclude that as h goes from h-, to p-, , arg n , ( h ) goes from a - n to a' - v . Similarly it may be shown that as h increases from p-l to A,, arg Ul(h ) increases from & a' - 7 to 4- a, so that it has increased by rr as h goes from A_, to A,. Generally, as h increases between two consecutive A,, or two consecutive p,, , arg n ( h ) increases by n. We can now find the variation in arg17,(h) as X describes the whole real axis. Suppose for definiteness that in the above table A, exists, to arg IT,(h) and so not & + p . Then as h increases from A, will increase by (m - l)n, in fact, with the previous determination of arg17,(h), we have
4
4
4
+
+
+
fr
4
4
1.8.
45
THE SECOND INVERSE PROBLEM
Also, as h increases to P,-~-, , the greatest of the two sets of eigenvalues in this case, the previous reasoning shows that arg 17,(X) will increase further to (1.8.8) arg fl1(pm-$-,) = $ a’ (m - p - I ) rr.
+
h
It remains to consider the variation of arg17,(h) when h
> pm-p-, . We write dl
= arg I7,(h-,)
< A,,
- arg n,(-m),
A , = arg f l , ( + ~) arg fll(pm-p-l).
+
Our aim is to show that A, A, = r - 8 (a’- a). Now this is certainly true (mod 2n),since the variation of arg I7,(X) round our semicircle is a multiple of 2n, the variation round the curved part of the semicircle is asymptotically mn, and that along the real axis from the lowest eigenvalue A_, to the greatest pm-p-l is (m- l ) +&(a’ ~ - a). We therefore have to dispose of the eventualities Al+A2=rr-’
2 (a’-
4+2P,
(1.8.9)
for some integral q # 0. Considering A , , as X varies in --oo < h < A_, , argn,(X) cannot reach either of the values +a‘ - pn, 8 a‘ - (p 1)n; as was shown above in considering the variation from A-, to ho , such values correspond to zeros of n4, that is to say, to members of the set p, , of which there are none in the interval (-m, LP). In view of (1.8.7) we deduce that
+
1 - %(a’ - a) < A ,
< 7r - +(a’
- a).
Similarly, in the interval (pm-p-l, +m), argn,(h) cannot reach the values Q a ( m - p - l)n, 8 a (m - p ) ~ since , these correspond to zeros of I f 3 that , is to say, to members of the set A,. Hence A , satisfies the same bound, and so
+
+
- (a’ - a )
< A,
+ A , < 2rr
-
(a’ - a).
+
Recalling that 0 < a < a‘ < 277, this implies that dl A, < 2n, which excludes (1.8.9) with q = 1, 2, ... . Similarly we exclude (1.8.9) with q = -1, -2, ... , so that in (1.8.9) q = 0. This completes the proof that the variation of argII,(A) is mn along the real axis, and so 2m7r around a large semicircle closed in the upper half-plane. Hence the zeros of IT, lie in the upper half-plane, and the ck given by (1.8.1), (1.8.5) have positive real part. T h e proof is similar in the case that is the greatest of all the 4 , pr and p - p the least.
1.
46
BOUNDARY PROBLEMS FOR RATIONAL FUNCTIONS
A few simple points complete the proof. Since I?,(O) = 1, D4(0)= 1, we deduce on putting A = 0 in (1.8.3-5) that I?,(O) = I?,(O) = 1. Thus Ill admits the expression ( 1 . 8 4 , in which the ck have positive real part. Also, on solving (1.8.34) for 17,,we get sin
+
(01’
- a) IT, =
-exp (--
+ id)sin (9
a)
n, + exp (-
+
ia)sin
(8or’) 17,.
Since Il, , 114 are polynomials with real coefficients, comparison with (1.8.5) shows that is the polynomial which is the complex conjugate to I?,, in the sense of having complex conjugate coefficients. Hence I?, is given by (1.8.1). Forming now ym(h)= DJI?,, we have the transfer function of the boundary problem, so that the recurrence relation (1.2.2) is determined, except as regards permutations among the c,, . We need, of course, to show that ym(h)= exp (ia) when A = A,, , and that y,(h) = exp (id) when h = p, . These follow immediately from (1.8.34).
n,
1.9. Moment Characterization of the Spectral Function This topic also belongs to some degree in the category of inverse problems. We focus attention on the dual orthogonality relations (1.4.12), and ask what other orthogonality relations there may be concerning the same rational functions ~ ~ ( with h ) possibly different A,, pr. T h e problem may be more compactly handled in the Stieltjes integral formulation (1.5.3). If we assume for simplicity that the eigenvalues are all finite, and write .(A) instead of T ~ & , ~ ( Athis ) , becomes (1.9.1)
Extending the definition of a spectral function given in Section 1.5 by actual construction, we may term .(A) a spectral function, associated with the recurrence relation (1.2.2), if it satisfies (1.9.1), together with some general restrictions. Here we shall require T(A) to be nondecreasing, right-continuous and to ensure absolute convergence in (1.9.1). T h e latter requirement is equivalent to (1.9.2)
An alternative formulation proceeds in terms of the eigenfunction
1.9.
47
MOMENT CHARACTERIZATION OF THE SPECTRAL FUNCTION
expansion, Theorem 1.4.6, in Stieltjes integral formulation (1.5.5-6). For arbitrary u, we define, as before,
and term .(A)
a spectral function if, whatever the choice of the u,, (1.9.3)
It is not hard to show that (1.9.1) implies (1.9.3), and conversely. The formulation in terms of the eigenfunction expansion has advantages for the corresponding problem for differential equations. We mention in passing that (1.9.1) does not ensure the validity of what has been termed the dual expansion theorem in which we start with any, suitably integrable, function v(h) and define the u, by (1.9.3), and then consider the expansion of o(h)in terms of ?,&(A), with the u, as Fourier coefficients. A spectral function with this additional property may be termed an orthogonal spectral function ; however, we shall not consider this here. The conditions (1.9.1), supposed to hold for 0 < j , K m - 1, constitute a moment problem, in that .(A) is to be found, as far as possible, from a knowledge of the “moments” with respect to it of the functions qj(h)qk(h).Just as moments of polynomials may be expressed in terms of moments of separate powers, so also the moments of the rational functions r/pjkmay be expressed in terms of moments of simpler functions of which they are linear combinations, namely, by resolving them into partial fractions. With this in mind, we define the function
m we suppose that the c, , n = 0, ..., m' - 1, have positive real part, and set up the boundary problem given by
= - +i,
n
..., m
= 0,
- 1.
(1.10.4)
supplemented in the case of s coincident c, by s - 1 equations of the form f'(i/cn) = ... =f(S-1)(i/cn) = 0. (1.10.5)
A more intrinsic formulation of (ii) is suggested by the observation that f(h) is a rational function that maps the upper and lower half-planes into each other; it is, in fact, the general such function which is finite at infinity. We may therefore pose (ii) as the problem of finding rational functions with these properties, which take the value - i at m specified points in the upper half-plane. We approach here the topic of the PickNevanlinna problem, though in restrictive fashion, as we confine attention to rational functions and require the interpolatory values all to be - 8;. The complete solution of these problems is given by:
+
Theorem 1.10.1, Let ~ ( hsatisfy ) either of the problems (i), (ii) above ; let it also be fixed so that ~ ( 0 = ) 0, and defined at points of discontinuity so that it is right-continuous. Let m' be the number of points of discontinuity of .(A). Then m' m. If m' = m, there is an a, 0 a < 27r,
>
0. Hence the circles C(b) given by (2.1.8) shrink progressively, or “nest,” in the sense that if b’ > b, then C(b‘) lies in the interior of C(b). In the case (2.1.6), y(b, A) diminishes to a positive limit, and, accordingly, C(b) shrinks towards a circle C(m), given by (2.1.8) with this limiting value of I y(b, A) 1 ; this may be termed the limit-circle case. If again (2.1.7) holds, so that 1 y(b, A) I + 0, then by (2.1.8) the circles C(b) converge on the point 1/(2i), so that we term this the limit-point case. T h e limit-point 1/(2i) is, of course, independent of the boundary parameter a, as was the limiting spectral function ~ ( h= ) h/(27~);the two are connected by the formal relation m
1/(2i) =
-m
(A - p)-I d
(5-),
though for a rigorous connection we have to use a more complicated integrand. Another aspect that discriminates between (2.1.6-7) is the existence of solutions of integrable square. T h e latter term is construed in conformity with the inner product appearing in the orthogonality relations (2.1.4). Since the equation (2.1.1) has essentially only one solution, y(x, A), the question is whether or for what h this solution satisfies (2.1.9)
It may be proved that this is certainly true if I m h > 0. For I m h = 0, it is true trivially if and only if (2.1.6) holds ;the same holds if Im h < 0. I n what follows, we prove that very similar statements hold for the recurrence relation (1.2.2) with rn = m.
2.2. Convergence of the Fundamental Solution We suppose defined an infinite sequence c,, , c1 , ... of possibly complex constants with positive real parts. T h e recurrence relation is, as before, n = 0, 1 , ... , (2.2.1) Y,,+~ = (1 ihc,,) (1 - ihE,,)-ly,,,
+
58
2.
THE INFINITE DISCRETE CASE
but with an infinite number of stages. For the fundamental solution we take yo = 1, defining, as before,
r-J n-1
y,(h) =
0
((1
+ ihc,) (1 - ihf,)-'>.
(2.2.2)
An immediate question is whether y J h ) tends to a limit as n + 00. We do not settle here this question in its full generality, but only in the case that the limit is to be meromorphic; this is equivalent to demanding that the zeros and poles should have no finite limit, which implies that c,--+O. Transferred to the unit circle, such questions are handled in greater generality in the theory of Blaschke products. In the following simple results we consider analogs of the criteria
(2.1.6-7).
Theorem 2.2.1. Let the constants co , cl, ..., have positive real parts. Then for y,(h) to tend as n-03 to a limit which is a meromorphic function of A, it is necessary and sufficient that and
C,
+O,
as
(2.2.3)
n +m,
(2.2.4)
where
a, = c,
.
(2.2.5)
if,
Let us first prove the conditions sufficient. We rewrite (2.2.2) in the form
provided that h # - i/& , p = 0, I, ...; these points have no finite limit, by (2.2.3).Provided again that h is not one of these points, 1/(1 - ihf,)
-+
1 as p
-+
00,
(2.2.6)
and hence by (2.2.4),
Thus the infinite product converges to a meromorphic function by standard tests.
2.2.
CONVERGENCE OF THE FUNDAMENTAL SOLUTION
59
Next we assume that yn(h) tends to a meromorphic function. By
(2.2.2), such a function would have to vanish at the points i/cD. Since y,(O) = 1, the function could not vanish identically, and so its zeros cannot have a finite limit. Hence (2.2.3) holds. T o prove (2.2.4) we use (1.4.3), where h is complex and not one of the points -i/Fp. We assume also that h is chosen so that I ym(h)I tends to a positive limit as m + m; this must be possible since yn,(0) = 1 for all m and since the limit is to be meromorphic. Choosing also A in the upper half-plane, we write (1.4.3) in the form
< l/{2 Im A),
(2.2.8)
Making m + 00, we get (2.2.9) Since I 1 - ihf, 1 1 as n-t 03, (2.2.3) having been proved already, and since by hypothesis I y,&(h)I tends to a positive limit, the convergence of the series on the left of (2.2.9) implies that of E r n f l , so that (2.2.4) holds. This completes the proof. We pass to the analog of the situation (2.1.7).
Theorem 2.2.2. Let the c, have positive real part, tend to zero as n --t 00, and let, with the notation (2.2.5), W
(2.2.10)
F u n = m.
Then if Im X > 0, yn(h)-+ 0 as n + 00, and if I m X < 0, Iyfl(A)] -+ m. If I m X > 0, the result (2.2.9) is available, and since c, --f 0 we have (2.2.11) Since Im h
> 0, R1 c,& > 0, we have by Lemma 1.2.1 that 1
= I yLl(h) I
> I ydh) I > I ydh) I > .*. 9
so that as n + 00, we have that I y,(h) I either tends to a positive limit or to zero. The former case is excluded as it would give a contradiction
2.
60
THE INFINITE DISCRETE CASE
between (2.2.10-1 1). Hence y,l(X) +0 if Im X > 0. If Lm h < 0, the fact that I y,(h) 1 -+ 00 follows from the case just proved, since
-
Y*(4 = l / Y d h
apart from singularities. Another aspect of the analogy between (2.2.1) and (21.1) which may be disposed of simply is that of “solutions of integrable square.” This phrase has here the interpretation that
$
an
I %(A) l2
0, and so there necessarily exists a solution of integrable square if Im X > 0, whether (2.2.4) holds or not. If (2.2.4) does hold, y,,(X) tends to a finite limit, and so (2.2.9) holds as a consequence of (2.2.7), apart from the poles h = -i& . I n connection with second-order difference and differential equations, a different pattern of results is encountered in connection with the existence of solutions of integrable square. In the present case the invariant form as given by (1.2.8) is definite.
2.3. Convergence of the Spectral Function We recall the definition (1.5.1-2) of r,,,(X) as a right-continuous step function, having jumps l/p, at the eigenvalues A,, and fixed by T,,,(O) = 0. Here A,, pr vary with m, as also with a. We consider here the convergence of r,,,(X) as m -+ 00, keeping for definiteness a fixed. This provides one of the main approaches to the proof of the eigenfunction expansion. A very simple, though somewhat vague, method of dealing with this question relies simply on the boundedness, uniformly in m and a, of the spectral function T ) ~ , ~ ( X ) The . boundedness of the spectral function may in turn be established in various ways, in particular by means of the dual orthogonality (1.5.3-4). T a k i n g j = 0 in the latter, we have
J
m
-m
I1
+
% l
dT,,,(X)
+ I c,
l-2Pm,a
=
I/% ;
(2.3.1)
2.3.
61
CONVERGENCE OF THE SPECTRAL FUNCTION
here the term p,,, occurs only if w is an eigenvalue of ( 1 . 2 . 2 3 , and in any event is nonnegative. Since Rl{c,} > 0, the function (1 h2)/ I 1 ihc, l2 is continuous on the real axis; it also tends to a positive limit as h + fw, and so has a positive lower bound c, say. Hence, for real A,
+
I
1
+ ihc,
1-2
+
+ P),
2 c/(l
and so from (2.3.1) we deduce the bound
f= dTm,cl(X)/(1 + h2) Q l/(cao),
(2.3.2)
-W
independently of m and a. The left-hand side is not less than
=
for any h
> 0, and so, for A > 0, 0 Q 7m,a(A)
Similarly, for any h
< 0, -(I
(1
+
X2)-1Tm,a(X),
< (1 + h2)/(cao)*
+ h2)/(cao)< Tm,,(h)
Q 0.
(2.3.3)
(2.3.4)
Since ~ , ~ . , ( his ) thus a uniformly bounded nondecreasing function, an application of the Helly-Bray theorem shows that we may choose , nonan m-sequence such that ~,,,(h) converges to a limit ~ ( h )also decreasing and satisfying the bounds (2.3.2-4). I n putting this result formally we incorporate the result concerning the passage to the limit for integrals involving the spectral function. Theorem 2.3.1 Let the c, , cl, ..., have positive real part. Then there exists at least one limiting spectral function ~ ( h )such , that for some m-sequence and any finite h we have Tm.,(h)
-
7(4*
(2.3.5)
This function is nondecreasing and such that (2.3.6)
There is a constant /3 3 0 such that for an arbitrary continuous function g(X), such that (1 h2)g(X) is uniformly bounded for all A, and such
+
62
2.
THE INFINITE DISCRETE CASE
+
that (1 A2)g(A) tends to the same finite limit go as A+ as m ---t 00 through the same sequence,
fm,
we have,
For the proof we refer to Appendix I.
2.4. Convergence of the Characteristic Function Closely linked with the convergence of the spectral function is that of, effectively, its Stieltjes transform, defined in (1.6.1). A curious feature is that in order to ascertain the boundedness and convergence of f,.,(X) for increasing m, it is advantageous to imbed this problem in the wider problem of the convergence of the set of values of fm,a(X), when a takes all real values, or indeed all values in the lower half-plane as well. This leads to the topic of nesting circles, which finds application in many of our boundary problems. Taking a real, it follows from (1.6.1) [cf. (2.1.8)] that Ifm.rr(h) - 1/(2i) I
= I ~ m ( h )I
Ifm.a(h)
+ 1/(2i) I.
(2.4.1)
As mentioned in connection with (2.1.8), for the analogous continuous case, this means that for fixed A, and varying real a, frn,.(A) 1’ies on a certain circle, C(m, A), say. Supposing that Im A > 0, so that I y,(A) I < 1, we can say that f,,,,(A) lies on the boundary of the finite disk D(m, A) of f-values characterized by
If-
1/(2i) I
< I Y m N I . If+ 1/(2i) I.
(2.4.2)
As m increases, ym(A)steadily decreases, if I m A > 0, and so these regions shrink, so that D(m, A) contains in its interior the disk D(m 1, A) and its boundary C(m 1, A). The conclusion may be drawn that either the circles contract to a point, in this case to the point 1/(22], or else to a limit-circle. A useful conclusion from the nesting-circle argument is that, whether the limit-circle or limit-point case holds, fin.a(A) is at any rate bounded, for fixed A with I m A > 0, independently of m and a. This gives an alternative proof of the boundedness of the spectral function. Putting A = i in (1.6.7), and comparing imaginary parts of both sides, all of which are negative, we deduce that
+
Pm.a
+ Irn dTm.a(P)/(l + P2) G I ~m {fm.a(i)) I. -W
+
(2.4.3)
2.5.
EIGENVALUES AND ORTHOGONALITY
63
Since the right-hand side is bounded, this yields bounds of the form (2.3.3-4). In passing to the limit as m d - in the partial fraction formula (1.6.2), or (1.6.6-7), it is necessary to modify the integrand so as to ensure absolute convergence. We rewrite (1.6.7) as
Here the real constant y k , a is bounded uniformly in m,a, by a similar argument to (2.4.3). Making m + a , we may assume that for some subsequence of m-values, convergence holds in &,,a, yk,a and ~ ~ , . ( p ) . By Theorem 2.3.1 we deduce that there holds a representation for the limit f ( X ) of the sequence of characteristic functions, of the form
2.5. Eigenvalues and Orthogonality I n this section we assume the constants c, to obey (2.2.3-5). The effect of this is that, as n --t ~ 3 , (2.5.1)
rI ((1 + ihc,) (1 - ihQ-l} W
yw(h) =
0
-
(2.5.2)
is meromorphic, with poles at the points l / ( i f p ) One . aspect of this is that the “limit-circle case” holds; making m + in (2.4.2), we see that if X is in the upper half-plane and is not one of the points i/cp, the characteristic function f is in the limit confined to a circle of positive radius. Another aspect is that eigenvalues and eigenfunctions can be defined, with very little difference from the finite-dimensional case discussed in the last chapter. For fixed real 01, we define the eigenvalues X, as the roots of the equation yw(h) = exp (icy). (2.5.3)
64
2.
T H E INFINITE DISCRETE CASE
As in Section 1.2, the reality of the eigenvalues follows from the fact that I ym(X)I < 1, > 1 for X in the upper or lower half-planes. Defining, analogously to (1.3.3) (2.5.4)
it is seen that each term in the latter sum increases by 2~ as X describes the real axis, so that O,(X) increases over an infinite range as X increases over the real axis. Hence (2.5.3) has an infinity of real roots. In the finite-dimensional case, m was admitted as an eigenvalue if it satisfied the determining equation (1.3.2). This does not in general apply to (2.5.3). The limit ym(-) for X tending to f 03 along the real axis will generally fail to exist, since y,(X) describes an arbitrarily large number of circuits of the unit circle as X describes the real axis. Instead, we consider whether (2.5.3) holds as h + im along the positive imaginary axis. Actually a further condition is required; we admit as an eigenvalue if, first, ym(4
as X+im
-
exp (4,
(2.5.5)
along the positive imaginary axis, and if at the same time, h2ym'(h)l{iym(X))
-+
P-*
(2.5.6)
for some /? > 0. This last condition, a modification of (1.4.9) for the point im, ensures that m as an eigenvalue corresponds to a positive weight in the spectral function. We may link up this eventuality of an infinite eigenvalue with the constants c, , on the one hand, and with the behavior of the characteristic function on the other. First we note: Theorem 2.5.1. In order that as X axis, we have (2.5.6) and also Ym(4
-
--f
im
along the positive imaginary
exp ( i 4
(2.5.7)
for some real a', it is necessary and sufficient that m
(2.5.8)
We first assume (2.5.6-7) and deduce (2.5.8). From (2.5.6) we have that ( d / d ) iogym(h) = O(h-'),
2.5.
EIGENVALUES AND ORTHOGONALITY
65
and so [logy,(h)]';"
=
O(h-1).
Using (2.5.7) we have that for large h on the positive imaginary axis, ym(h)= exp (ict')
+ O(h-l).
(2.5.9)
Next we use (2.2.7). Making m - + m we have, with a slight rearrangement,
If we make h -+ along the imaginary axis in the positive sense, the right-hand side remains bounded, by (2.5.9). Hence the left-hand side of (2.5.10) is also uniformly bounded. Making h -+im in the individual terms on the left, we have I y,(h) I -+ 1, and Q)
I h/(l - ihf,,) I -,1/1 cn 1. Applying this limiting process to the sum over 0 making m + Q),we derive
< n < m and then
m
(2.5.1 1)
in partial verification of (2.5.8). We write provisionally (2.5.12)
By logarithmic differentiation of (2.5.2), we have y,'(h)/y,(h)
=i
2 an(1 + ihcn)-' (1 - ihfn)-l, 0
the series converging absolutely at least for purely imaginary A, and so the left of (2.5.6) admits the expression m
h2ya'(h)/{iym(X)}= z u n I cn 0
(1
1
(2.5.13) + =)(1 - A)-'. axe, -1
2.
66
THE INFINITE DISCRETE CASE
We wish to make h + im in the factors on the right. T o justify this we note that (2.5.11) may be written in the form
2 R1{cn}/l c,, l2 < W
2
00,
so
0
$
that
R1{cn}/l c,
I < 00,
where we use the fact that c, + 0. It follows that for n > no , say, either + r > arg c, > a r , or else - i r < arg c, < - 7. Hence if A is a pure imaginary, either * T < arg (ihc,) < $ r , or else - $7 < arg (ihc,) < - a n ; the same bounds apply to arg (At?,), if h is purely imaginary and n > n o . It follows that, under the same circumstances,
a
I 1 + l/(ihcn) I > 2-1'2,
I 1 - 1/(iMn) I > 2-lI2.
(2.5.14)
We deduce that the series on the right of (2.5.13) is uniformly convergent. Making A -+ im in the individual terms on the right, and using (2.5.12), we get h2ym'(h)/{iyw(X)} 1/F, +
and on comparison with (2.5.6) we have 8' = 8, completing the proof of the necessity. Next assume that (2.5.8) holds. By the argument just given we deduce (2.5.6). It remains to prove (2.5.7). It follows from (2.5.6) that (d/dh)log yw(h)= O(h-2),so that ym(h)tends to a limit as h -+im. In order to establish (2.5.7) for some real a', it will be sufficient to show that I yw(X)I + 1 as h + im, still along the positive imaginary axis. To do this we use (2.5.10) again, writing the left-hand side in the form m
By (2.5.14), the last factors are uniformly bounded, and also we have I y,(h) I < 1. Hence this series is uniformly bounded. Hence the right of (2.5.10) is uniformly bounded, whence it follows that 1 - Iym(h)
1'
=z
O(h-'),
which completes the proof. T o set up a connection between this case of an infinite eigenvalue and the behavior of the characteristic function, we have: Theorem 2.5.2. Let (2.5.5-6) hold. Then, as h+im positive imaginary axis, fm.dh) -PA-
-
atong the (2.5.15)
2.6.
ORTHOGONALITY AND EXPANSION THEOREM
67
As in the proof of (2.5.9), we deduce from (2.5.5-6) that
-
[ l o g ~ m ( ~ ) Ii/(PA)j ~~
whence
-
exp (ia)- ym(h)
Substituting this in the expression fm.a(A) =
i(Ph)-l exp (ia).
+
(2iFYexp (ia) Ym(h)) { ~ X(;a) P - Ym(A))-l,
we deduce (2.5.15). This justifies the use of the notation /lfor the various quantities in Sections 2.3, 2.4, and 2.5.
2.6. Orthogonality and Expansion Theorem If we confine attention to the meromorphic case in which there exist eigenvalues h, , the roots of (2.5.3), with no finite limit-point, we obtain orthogonality relations very similar to those of Section 1.4. The eigenfunctions are now infinite sequences Yo(Ar), Yl(hr),
9
which for the purposes of orthogonality have to be modified to 710(Ar),
?l(M>
' I *
f
(2.6.1)
where r],(A) is given as before by (1.4.5-6). The latter are orthogonal in a similar manner to (1.4.7) in that (2.6.2)
For the proof, we take h = A, , p = A, in (1.4.1) and make m * 00. Subject to the assumptions (2.2.3-4) of the meromorphic case, the lefthand side of (1.4.1) tends to a limit as m --+ m, which is zero by the boundary condition (2.5.3). T h e conditions (2.2.3-4) also ensure absolute convergence in (2.6.2), in view of (1.4.10). Parallel to (1.4.8-9), we have the normalization relations (2.6.3)
as in (2.5.13).
68
2.
THE INFINITE DISCRETE CASE
If 00 is an eigenvalue, according to (2.5.5-6), we have in addition a corresponding eigenfunction, the sequence Tot, T l t ,
(2.6.6)
8
where Tnt is as defined in (1.4.18). These are orthogonal to the sequences (2.6.1) in that (2.6.7) n=O
The proof consists in taking t+ = X, in (1.4.1), with h on the positive imaginary axis, making first m -+ and then h -+.,i The first process gives
We get (2.6.7) formally on making A + ,i in view of (1.4.18). This limiting process may be justified by uniform convergence. Since c, + 0, by (2.2.3), we have from (1.4.10) that Tn(h) = O(1).
From (1.4.5) we have, since 1 y,(h)
I < 1 when I m h > 0,
I yn(A) I < I 1 - ihfn 1-l < I iACn since - ih is real and positive and R1 c,
I-l,
> 0. Hence
(A - A,) Tn(A) = O(l/l cn
I)*
Hence the series (2.6.8) will be uniformly convergent, for A on the positive imaginary axis, if
is absolutely convergent. This follows, using the Cauchy inequality, from (2.2.4) and (2.5.8); the latter was found to be necessary for 00 to be an eigenvalue. Finally, there is the normalization relation
This is the same as (2.5.8).
2.6.
ORTHOGONALITY AND EXPANSION THEOREM
69
Next there will be a second set of orthogonality relations, dual to (2.6.2), (2.6.3), and parallel to (1.4.12). In the finite-dimensional case of Section 1.4, these dual relations were deduced as a direct consequence of the orthogonality of the eigenfunctions. In the present infinitedimensional case such a deduction is no longer possible. Instead, we use the method of limiting transition as m -+ m from the finite-dimensional case. We make m -+ 00 in (1.5.3). Since yr,&(A)tends to a meromorphic limit as m -+ 03, the eigenvalues A,,, of the finite-dimensional problem and the normalization constants P , , , ~ ,will tend as m -+ 00 to the corresponding quantities for the infinite problem, the A, as given by (2.5.3) and the p, as given by (2.6.5). Hence the spectral function T,,,~(A) will tend to a limit ~,,,(h) defined by (2.6.10) (2.6.1 1)
Making m
-+ 00
in (1.5,3), and using Theorem 2.3.1, we get (2.6.12)
for j , K = 0, 1, 2, ... . The term in /3 is to be omitted unless m is an eigenvalue, that is to say, unless (2.5.5-6) hold, and so also (2.5.8). The eigenfunction expansion is an immediate consequence of these dual orthogonality relations (see Appendix 111). In conformity with the orthogonality (2.6.2), it relates to arbitrary infinite sequences (2.6.13)
(2.6.14)
We write (2.6.15)
so that ~ ( 4is) the Fourier coefficient of (2.6.13) with respect to the eigenfunction (2.6.1); the series will be absolutely convergent for real A
70
2.
THE INFINITE DISCRETE CASE
by (2.6.14), (2.2.4), and (1.4.10), in view of (2.2.3). For the event of an infinite eigenvalue we define likewise
3anun m
ot =
qnt.
(2.6.16)
The eigenfunction expansion is then (2.6.17) (2.6.18)
where the term in
fl is omitted if
a3
is not an eigenvalue.
2.7. A Continuous Spectrum In the case in which (2.2.3) holds but not (2.2.4), we have the limitpoint case in which y,(X) +0 as n -+ a3 for h in the upper half-plane. In this case the direct orthogonality (2.6.2) fails. The eigenvalues cannot be defined and the series (2.6.2) will not be absolutely convergent, nor will the series in (2.6.3). Nevertheless, the dual orthogonality of the type of (2.6.12) remains in force; we remark in passing that the opposite situation prevails for (2.1.1), in that the eigenfunctions are orthogonal by (2.1.4), but the dual orthogonality has no sense. To adapt (2.6.12) to the present case, we take .(A) = X/(27), fl = 0, and the formulas in question are
These formulas are true with the sole assumption that the c , ~ have positive real part; they may be verified directly, for example, by contour integration, using the expressions (1-4.lo), (1.9.10-1 1). An expansion theorem may be deduced. For example, if in the sequence (2.6.13) all members beyond some point are zero, and .(A) is defined by (2.6.15), then un =
J”
m -m
~ ( hvn(h) ) d[h/(2r)1-
(2.7.2)
For the proof we need only substitute for w(h) and integrate term by
2.8.
MOMENT AND INTERPOLATION PROBLEM
71
term, using (2.7. l), the integrals converging absolutely. The restriction that the sequence (2.6.13) vanish beyond some point is clearly too severe, but we shall not investigate this further here.
2.8. Moment and Interpolation Problem Of these various questions of inverse type raised in Sections 1.7-1.10, we take up here the determination of the spectral function by the orthogonality. The property postulated is that for some nondecreasing rightcontinuous function .(A), of bounded variation in any finite interval and such that, in addition to ~ ( 0 = ) 0, (2.8.1)
and for some constant
I
m
-cc
p >, 0, we are to have
a +
vg(~)
dT(x)
&g+
rlKt = S,,/ai
9
(2.8.2)
for all j , k = 0, 1, ... . Here qj(A), qjt are given as before by (1.4.6), (1.4.18), for a given set of constants c,, with positive real part, while ,8 does not necessarily have the value given by (2.5.8). As we have seen, the problem of finding such ~ ( h?!)t , has at any rate one solution, given ) A/(27r), ,8 = 0. by ~ ( h= The problem may be considered as a moment problem, “determinate” if the solution is unique, namely, that just mentioned; and “indeterminate” if there is more than one, and so an infinity of solutions. As in Section 1.9, a first step is to replace this by a simpler moment problem. In modification of (1.9.4), we define now (2.8.3) (2.8.4)
this modification being necessary to ensure absolute convergence in (2.8.34) on the basis of (2.8.1). Here y is an indeterminate real quantity. We can then assert that it is sufficient for (2.8.2) that f(i/c,) = - +i,
p
= 0,1,2,
... ,
(2.8.5)
if the cp are all distinct ; if they are not all distinct, and s of them coincide
2.
72
THE INFINITE DISCRETE CASE
in some value c, we are to have s - 1 additional differentiated equations, namely, f’(i/c) = f”(i/c) = ... =f(-(i/c) = 0. (2.8.6) Let us for example deduce (2.8.2) with j parts in (2.8.3), h being complex, we get Im { f ( A ) }
=
-1m { A }
J
--a,
= k.
dT(p)/I A - p
On taking imaginary
l2 - /3 Im {A}.
(2.8.7)
Putting h = i / c j we have, in view of (2.8.5),
or 1 =(cj+f5)IS
m --a,
d+Il
-~p~jI-2+BIc~l-z/,
which is equivalent to (2.8.2) with j = k, by (1,4.10), (1.4.18), and (2.2.5). We rewrite the result to be proved, (2.8.2), to deal with the case j # k, in the form
J
m --a,
dT(CL)77dd
77ko + B ?ifnm PZ775(CL)6x4 = 0.
(2.8.8)
Supposing for definiteness that j < k, and for simplicity that the c, are all distinct, we have by (1.9.7) the partial fraction representation, for real p,
where k
Z d . = 0, 3
and so, by an easy calculation,
The result (2.8.8) to be proved then assumes the form
(2.8.9)
2.8.
73
MOMENT AND INTERPOLATION PROBLEM
This follows on taking linear combinations of the results (2.8.5) and using (2.8.9). We omit the details of the calculations for the event that the c, are not all distinct, as also the proof that the conditions (2.8.5-6) are necessary for (2.8.2);these are closely similar to arguments given in Section 1.9. T h e conclusion is that (2.8.2) may be replaced by (2.8.5-6), which is a moment problem also, in which we consider the moments of the elementary functions ( A - I.)-', h = i / c p , p = 0, 1, ... . A still simpler formulation is obtained if we observe that (2.8.4) is the general expression of a function ,f(h), which is regular in I m A > 0, and satisfies there Imf(X) 0. We are now asking for a function of this class with the interpolatory properties (2.8.5-6). As previously mentioned, this falls within the topic of the Pick-Nevanlinna problem. For its solution, we impose to begin with m of the conditions (2.8.5), with such of (2.8.6) as may be relevant in the event of c o , ... , c,,,-~ not being all distinct. T h e function
0 and so y,(h) as fixed with 1 ym(X)1 < 1, we may treat #(A) as disposable subject to I #(A) I 1. I n the form (2.8.10) this means that
0, f(h) must tend to a unique limit, in fact to - &i.
2.9. A Mixed Boundary Problem As a check on the generality of the recurrence relation (2.2.1) as a source of boundary problems, we apply the procedure of the “first inverse spectral problem” of Section 1.7, in which we start with the spectral function and boundary condition, and attempt to recover the recurrence relation. In the situation of Sections 2.5-6, when the infinite recurrence relation yielded a meromorphic limit, the spectral function was a nondecreasirig step function ~ ( h )whose , points of increase have no finite limit-point, satisfying the boundedness condition (2.3.6); there was also a constant /I 2 0, which could be considered as a jump of ~ ( hat) infinity. The question is then whether, given such a .(A) and a /I 2 0, and a real a prescribing the boundary condition, we can find ) to (2.6.10-1 l), and a recurrence relation yielding this ~ ( h according also /I corresponding to an infinite eigenvalue, if /I > 0. Solving this problem according to the method of Section 1.7, we form the characteristic function f(h) according to (2.8.34), and y(X) according to
+ l}.
y(h) = exp (ia.){2if(h) - 1}/{2if(h)
(2.9.1)
Here y in (2.8.3-4) is to be fixed so that y(0) = 1 ; as in Section 1.7, y is fixed uniquely if exp (ici)# 1 and h = 0 is not a point of increase of ~ ( h )and , in the contrary case is arbitrary, subject to being real. The problem is then to factorize y(h) in the form (2.5.2). Considering the general nature of these functions, f(h) as given by (2.8.34) is meromorphic, with its only poles on the real axis, and mapping the upper and lower half-planes into each other; it is indeed the general such function. By (2.9.1), y(h) will be meromorphic, and satisfy I y(h) I < 1 if Im A > 0, I y(h) I > 1 if Im h < 0, and I y(X) 1 = 1 if Im h = 0. In addition, we are to arrange that y(0) -- 1. However if we adopt these as the requirements for y(h), to be the transfer function of some recurrence relation, we find that y(h) need not be representable in the form (2.5.2), or (1.3.1), since the function exp (ich), c > 0,
2.9.
75
A MIXED BOUNDARY PROBLEM
also fulfills these conditions. T h e general function with these requirements is, in fact, given by a combination of the two types, namely (2.9.2)
subject to the conditions c
Rl{c,}
> 0,
real,
c
2 0,
2 Rl{c,}
0, when (2.9.2) reduces to an exponential only, m finite, in which case the latter conditions in (2.9.4) are unnecessary, and m infinite. Corresponding to (2.9.2), we can form the mixed differential recurrence relation for a function y(x, A), defined continuously in - c x 0, and discretely for x = 0, 1, 2, ... , m, by
<
1. Hence
+
and the result (2.10.1) follows on summing over Y = I' 1, ... . The proof of (2.10.2) is entirely similar. The next result plays a similar role to those, such as the RiemannLebesgue lemma, asserting that if a function is sufficiently smooth, then its Fourier coefficients tend to zero with a certain rapidity.
Lemma 2.10.2. n-1
Writing
+
n
~ ~ (= hn ) (1 ihc,) n ( 1 - iAc,)-l, 0
n
...,m
= 0,
- 1,
(2.10.5)
0
we have, for real A and arbitrary u o , ...,
For the proof, we have with the notation (2.10.7)
78
2.
THE INFINITE DISCRETE CASE
=
-2iXcn
~
-)ln(X).
Hence
and the result follows on taking absolute values, noting that 1 y,(A) 1 = 1 for real A. T h e expansion theorem to be established is, formally, that for given u(x), -c x 0, x = 1, 2, ..., we definc
<
0. For any positive integer s we form an approximating recurrence relation with 2s stages based on real constants c:), ..., czs-l (S) , where cb"' = c y = c(a)
= co
... =
CS"il =
c (8-1 8)
c1
,
(2.10.12)
= c/(a),
...,
= '8-1
'za-1 (a)
(2.10.13)
Here co , c l , ... are the constants appearing in the boundary problem (2.9.5-9), assumed real and to satisfy (2.9.4), save that if m is finite, c, = c,+~ = ... = 0. Writing (2.10.14)
we have (2.10.15)
+
(4
Since [l iAc/(2s]8 + exp iAc) as s -+ 03, we have, on comparison with (2.9.2), for real y(*)(A)
-+
c,
y(h) as s -+ 00,
, that (2.10.16)
uniformly in any A-region of the form
c1
where is arbitrary, and & is such that this region does not include any of the poles -i/c? . Next we set up eigenvalues and normalization constants for the approximating problem, observing that these tend to the corresponding quantities for the problem (2.9.5-9). We denote by A:) the roots of y("(h) = exp (ia),
(2.10.18)
identifying them as in Lemma 2.10.1. Since (2.10.16) holds uniformly in any region (2.10.17), we have A:)
-+
A,
as
s
-03,
(2.10.19)
h, being the root of (2.9.9), which is identified in the same manner; the latter are, of course, simple roots of (2.9.9) in view of (2.9.16-17).
80
2.
THE INFINITE DISCRETE CASE
Similarly, the normalization constants p y ) associated with A?) will, by (1.4.9), be given by =~ ( S ) ' ( A ~ ) ) / { ~ ( S ) ( A ~ ) ) , (2.10.20) and on comparison with (2.9.17), we see that pp)-+pr,
as
s+w.
(2.10.21)
Here we rely on the convergence (2.10.16) in the sense of the uniform convergence of analytic functions. We may now set up the Parseval equality for the approximating problem and carry out the limiting transition. We write (A) =
n--l
(8)
vn
so that, if n
< s,
7:)(A)
=
[l
r=O
(1 + iAcg"')
n n
(1 - i A C y ,
(2.10.22)
0
+ iAc/(2s)]"[l - i A c / ( 2 ~ ) ] - ~ - '
-+
exp (iAcn/s) (2.10.23)
as s ---t 03, where n may vary with s, uniformly in the A-region (2.10.17). If n 2 s, say, n = s t , then
+
t-1
7p(~) -+exp ( ~ Ac ) r-0
(1 + ~AC,)
IT(1 t
~ACJ-~.
r=o
(2.10.24)
We define, for the approximating problem, w(s)(A) =
2 (c/s) s-1
U( -C
+~c/s)
+ 2 atu(t)7zt(A). 8-1
(2.10.25)
If we use (2.10.23-24) and make s -+w,the first sum becomes an integral, the second, in general, an infinite series, and so we get, on comparison with (2.10.8), d 8 ) ( h )-+ v(A), as s+ 00. (2.10.26) I n considering the limiting transition as applied to the second sum in (2.10.25), we use the fact that Z 2 t < w, that I u(t) 1 is bounded, by (2.10.11), and that, in the present case I rg)(A)1 1 if A is real. The finite-dimensional Parseval equality then assures us that, by (1.4.15),
dx
+ (hA.L)-l j
0
-1
Y ( X , hr)Y(X,
A,) dx = 0. (2.11.4)
We shall not take up these problems here. The one just mentioned may serve as representative of those for which the transfer function, in this case exp ( i A - i / A ) , maps the upper and lower half-planes into the interior and exterior of the unit circle, being analytic almost everywhere on the real axis. In a more general class of problem, the transfer function will merely map the upper half-plane into the interior of the unit circle.
CHAPTER 3
Discrete Linear Problems
3.1. Problems Linear in the Parameter In the previous two chapters we studied a recurrence relation (1.2.2) connecting successive complex numbers yn , ynfl with the property of preserving length, I y, I = I yn+l.I, for real parameter values. This relation was in general, of necessity, bilinear in the parameter A. If we move on to higher-dimensional cases, yn being a vector, a simpler possibility presents itself, namely that the recurrence relation is linear in the parameter. Although the more general bilinear relation can be studied also in the matrix case, the linear one is of special interest, and is the subject of the next four chapters, with reference to orthogonal polynomials. The recurrence relation in its general form will be yn+1 = (AAn
+ Bn)yn
9
ft
= 0,1,
9
(3.1.1)
where yT1is a vector, or rather a k-by-1 column matrix, and the A,, B, are k-by-K matrices. In generalization of the length-preserving property of (1.2.2), we postulate that for some fixed k-by-k matrix J and for some suitable A-set we are to have
JYn+l
~n+i*
= Yn* JYn
(3.1.2)
9
for all n, where the (*) indicates the complex conjugate transpose. We take J to be nonsingular and symmetric or else skew symmetric, in the real or Hermitean sense, but do not restrict it to be positivedefinite. As a suitable A-set we shall admit either the real axis or the unit circle; we do not wish that (3.1.2) should be a consequence of (3.1.1) for all A. That circles or straight lines are the appropriate curves may be seen by setting up sufficient conditions for (3.1.2) to hold. Substituting from (3.1.1), we get
+
~n*(hAn B n ) * l ( W 83
+ Bn)Yn =
~ n * l 1~ n
84
3.
DISCRETE LINEAR PROBLEMS
which will be so for any yn if
(un* +
J W n + &)
=
I.
(3.1.3)
If this equation be written out in terms of the entries of the matrices, we should get k2 equations of the form ahX
+ b,h + b,X + c = 0,
which, if representing any curves, represent circles or straight lines.
A linear transformation in X can transform the curve in question into
the unit circle or the real axis, as the case may be. To complete the boundary problem in the finite discrete case we suppose that the recurrence relation (3.1.1) is defined for n = 0, ..., m - 1, yielding a sequence of k-vectors y o , ...,y,rL, which is determinate when the first is known. We impose boundary conditions on y o , ym of the following form. There are prescribed boundary matrices M , N , square and of the k-th order, and subject to M* JM
= N* JN ;
(3.1.4)
they are also to have no common null-vectors, i. e. Mv = Nv = 0, v a column vector, must imply a = 0. We ask for solutions of (3.1.1) such that there exists a column matrix v # 0 with the property that y m = Nv.
y o = Mv,
(3.1.5)
The requirement that M , N have no common null-vector ensures that at least one of y o , ym does not vanish, in the sense of the vanishing of all their entries. By the recurrence relation, ym # 0 implies yo # 0, so that in any event y o # 0. The requirement (3.1,4),together with (3.1.5), may be thought of as requiring that y o , ynr should be of the same “length,” in the sense that YO*JYO=Ym*JYn * (3.1.6) For by (3.1.5) this is equivalent to v*M* JMv
= v*N* JNv,
which is, of course, implied by (3.1.4). We may express this by saying that the sequence of mappings (3.1.1) form, when applied successively and starting with the particular vector y o , an “isometry,” in that (Urn-1
+
&n-l)
(hA0
+ &)Yo
3.2.
85
REDUCTION TO CANONICAL FORM
has the same length as y o . With certain additional restrictions, the conclusion can be drawn that each of the separate mappings (3.1.1) is an isometry, and that A, an eigenvalue, must lie on the real axis, or the unit circle as the case may be.
3.2. Reduction to Canonical Form Without loss of generality, the matrix J characterizing the invariant quadratic form may be supposed to have one of certain special forms, in particular, a diagonal form made up of 1’s and -1’s. To make such a reduction, suppose that J is related to a Jo of some special form by J
=
(3.2.1)
K*JoK
where K is nonsingular. If we define KYn
= Y nt ,
(3.2.2)
the invariance relation (3.1.2) becomes another of the same form, (3.2.3)
The recurrence relation becomes, in terms of the y i
, (3.2.4)
and the boundary conditions are now that, for some v # 0, y,J = KMv,
yi
= KNv.
(3.2.5)
The new boundary matrices K M , K N satisfy, by (3.1.4) and (3.2.1),
which correspond in form to (3.1.4). T h u s the boundary problem has been transformed to one of the same type, with Jo for J. Suppose that J is Hermitean, and as always nonsingular. I n this case we may connect J with a Jo according to (3.2.1), where Jo is diagonal, with diagonal entries 1, - 1 corresponding to the positive and negative
86
3.
DISCRETE LINEAR PROBLEMS
eigenvalues of the matrix J. If J has p positive and q negative eigenvalues, p q = k, we may take loin the form
+
(3.2.7)
where Ep , E, are the pth order and qth order unit matrices, the matrix
Jo being completed with zeros. Another important case is that in which
J is real and skew symmetric; since J is to be nonsingular, even. In this case we may take
K must be (3.2.8)
Having arranged that J should be in a suitable form, we may next standardize the form of the recurrence relation. If now we write, for some nonsingular H,, , HnYn == zn i (3.2.9) the recurrence relation becomes (3.2.10)
and the boundary conditions are z0 = H0Mv,
Z,
= H,Nv.
(3.2.11)
Provided that the H , satisfy Hn*JHn = J,
(3-2.12)
it may be verified that the new problem (3.2.10-1 1) has the same properties as the old one, in that z,*Jz,, is independent of n, for X real or on the unit circle as the case may be, while (3.1.4) holds with HoM, H,N for MI N . We can then choose the H , successively so as to make the right-hand side of (32.10) have some special form; the details will, of course, vary from case to case. We shall have much occasion to consider the set of matrices 2 such that, for some fixed J, (3.2.13) Z*JZ= J. I n the special case when J = E, the unit matrix, these form the set of unitary matrices. In the general case we may term them ]-unitary. It will always be the case that J is nonsingular, in which case the J-unitary matrices form a group, the group U(J) say. In the case (3.2.8) this group is sometimes termed the symplectic group.
3.3.
87
THE REAL AXIS CASE
3.3. The Real Axis Case We now give the form of some boundary problems satisfying the restrictions of Section 3.1. We take in this section the case in which the quadratic J-form is invariant for A on the real axis. We first find some recurrence relations which have this property. Taking a general recurrence relation as typified by the transformation
the property required is that if a second column matrix z is transformed
then z*Jy is unchanged, provided that A is real. This is equivalent to
(M* + B * ) J(u +B)
=
J
(3.3.3)
+
for all real A, and indeed for all A; in other words, AA B is to be J-unitary for real A. Comparing powers of A in (3.3.3), we wish to find matrices A, B satisfying B*JB = J , (3.3.4) A* JB
+ B*]A = 0, A*JA
If we write (AA
+ B)
=
(AAo
(3.3.5)
= 0.
(3.3.6)
+ E ) B,
(3.3.7)
where B is J-unitary according to (3.3.4), it will be sufficient to ensure that AA, E is J-unitary for real A. For this case (3.3.5-6) become
+
A,* J
+ JA, = 0,
A,* JA,
= 0.
(3.3.8)
We now take J to be skew-Hermitean, so that J*
=
-1.
(3.3.9)
Then the first of (3.3.8) may be written
( P o ) = (JAo)*, so that if we define JA, = C, we may replace (3.3.8) by C
=
C*,
C*J-lC
= 0.
(3.3.10)
88
3.
DISCRETE LINEAR PROBLEMS
Thus suitable matrices A, B are given by XA
+ B = (XJ-lC + E ) B,
(3.3.11)
where B is ]-unitary, C is symmetric in the Hermitean sense, and satisfies the second of the equations (3.3.10). Our problem is thus reduced to finding Hermitean matrices C such that C*J-'C = 0. If J has all its eigenvalues of the same sign, in the imaginary sense, that is to say, if J/i, which has its eigenvalues real, is positive-definite or negative-definite, this problem has only the trivial solution C = 0. For if say J/i> 0, then J-'i > 0, and so C*(J-'i) C 2 0 if C #O, in the sense that C*(J-'i)C = 0 is excluded. Suppose then that J has (imaginary) eigenvalues of both signs; this is assured if k is even and J is real and skew symmetric. In this case (3.3.10) has nontrivial solutions. Let p >, 1 be an integer such that J has at least p eigenvalues of both signs. It may be shown that there exists an "isotropic" set of p vectors tl, ..., t p ,that is to say, column matrices, such that (3.3.12) [,* J-'ts = 0, I , s = 1 , ...,9. If then we put (3.3.13)
we have which vanishes by (3.3.12). If, in addition, we impose on the numerical coefficients yrs the symmetrical conditions f r s = ysr
r , s = 1,
-*.,P,
(3.3.14)
then C will be Hermitean. Finally, for later purposes it will be necessary to impose a definiteness condition on C, restricting its sign; if we require that C 3 0, this will be ensured by imposing a similar condition on the yra 9 namely, (3.3.1 5)
So far as the restrictions M*JM = N*JN are concerned, special interest attaches to the case in which M*]M
= 0,
N*JN
= 0.
(3.3.16)
3.4. THE
89
UNIT CIRCLE CASE
This again is impossible if J , or J/i,is positive or negative definite. Suppose however that k is even, to take a simple case, and that the eigenvalues are & k of each sign, so that there exists an isotropic set of p = & k vectors satisfying (3.3.12). Here we need only take M , N to have the form
where u l ,
..., u k
are 1inear.ly independent column matrices.
3.4. The Unit Circle Case We pass to the other case in which z*Jy is to be invariant under (3.3.1-2) provided that [ h I = 1. We must now have (XA*
if
IhI
+ B*) J(hA + B ) = J
= 1, and so, since then
(X-lA*
=
A-l,
+ B*) J ( M + B ) = J ,
(3.4.1)
for h on the unit circle, and indeed for all A. Comparing coefficients we see that it is necessary and sufficient for the required invariance that B*JA = A*JB A*JA
(3.4.2)
= 0,
+ B*JB = J.
(3.4.3)
We may now conveniently take J to be Hermitean instead of skewHermitean; it is permissible that J be positive definite, though this is not the most interesting case, since it is more restrictive on the boundary conditions. To construct solutions of (3.4.2-3), we suppose for definiteness that J has some positive eigenvalues, and select some number p of them. Let y l , ..., y p be these eigenvalues, tl, ..., tp the corresponding eigenvectors; we number the remaining eigenvectors from p 1 to k. If then we take
+
then (3.4.2) will be satisfied whatever the values of the numerical coefficients a T 8 ,&; here we suppose the C1, ..., (k orthonormalized.
3.
90
DISCRETE LINEAR PROBLEMS
Since
J
k
=
Lr&*
1
we may break up (3.4.3) into two separate equations as (3.4.5) (3.4.6)
T o write the latter in terms of the coefficients ars, jlrs, let A, denote the p-by-p matrix of the a r s , B, the (K - p)th order matrix of the &, , r, the diagonal matrix with entries y1 , ..., yp , P, the diagonal matrix of the y p f l , ..., y k . Then (3.4.5-6) are equivalent to A,*r,A, B,*r,B,
r, , = r, .
(3.4.7)
=
(3.4.8)
Having found particular solutions of (3.4.2), (3.4.3), that is to say, a B which is ]-unitary for h on the unit circle, further particular XA cases may be found by multiplying, on either side, by an arbitrary constant ]-unitary matrix.
+
3.5. The Real 2-by-2 Case In this, which is substantially the case of ordinary orthogonal polynomials, we take
J
=
(y -3
(3.5.1)
and seek real 2-by-2 matrices A, B such that for arbitrary real 2-vectors
y, z if
then
yt
= (XA
+B)y, ~ tJyt *
ZT = (XA = z*]Y,
+ B) I,
(3.5.2) (3.5.3)
whenever h is real. We find here a heuristic solution, which is substantially the general solution, though we shall not prove this.
3.5. THE
REAL
2-BY-2
91
CASE
With a suitable convention as to sign, z* Jy represents the area of the parallelogram with y , z as two of its sides. Hence our requirement is that the matrix h A B, interpreted as a mapping of the plane into itself, should leave area unchanged. This may also be seen in another way. For a real 2-by-2 matrix to be symplectic, it is necessary and sufficient that it be unimodular, that is, have determinant unity. This again implies the invariance of area. Our quest is therefore for mappings, linearly dependent on a real parameter A, which preserve area. I n geometrical language, such a transformation is given by a “shear,” or a “symplectic transvection.” T o form such a transformation, we take a fixed line I in the plane, for any point P in the plane drop a perpendicular PQ to I, and form the transformed point P’ by moving P a distance (ha P)PQ parallel to I ; regard is to be had to the sense of the motion and to the sense of PQ. I t is easily seen that this transformation leaves area unchanged. Another transformation leaving area unchanged is rotation about a point, and we may form a boundary problem by imposing on P a succession of transformations of the above form, relative to a set of concurrent lines I,, interspersed with rotations. T h e boundary conditions may require that P start and finish on some line, for example. For the standard form of such transformations, we take the shear
+
+
x1-
x1
+ (ha + B)
x2
,
x2
-
,
x,
(3.5.4)
where xl, x2 may be thought of as coordinates, succeeded by the “rotation” x1-
-
-x2
,
x,
-
-
x1
.
(3.5.5)
The combined transformation may be written x1
-x2
,
x2
x1
+ (ha + B) -
(3.5.6)
x2
For the recurrence relation we take a series of such transformations, obtaining Xl.n+l
=
-%.n
I
XZ.n+l
= x1.n
+ (horn + B J
x2.n *
(3.5.7)
On substitution from the first equation into the second, we have Xz.n+l == -Xz.n-1
+ (horn +
Bn) x2.n
3
(3.5.8)
which forms a three-term recurrence relation, which we treat as the origin of orthogonal polynomials.
92
3.
DISCRETE LINEAR PROBLEMS
3.6. The 2-by-2 Unit Circle Case This leads to the topic of orthogonal polynomials on the unit circle, and to another case in addition, There are here two distinct possibilities for the matrix J characterizing the quadratic form. In the first of these we take (3.6.1)
which is similar to (3.5.1) and not essentially distinct from it. Again we ask that the 2-by-2 matrices A, B should have the invariance property (3.5.2-3), with J as given by (3.6.1) and for arbitrary h on the unit circle; in general the property is not to hold for h off the unit circle. Special cases of such pairs A, B are given by +
=
A 0 1 0 A 0 (0 1)’ (0 A)’ (0 A).
(3.6.2)
From these further cases may be constructed by multiplying by a general matrix A’ which is J-unitary, that is, A’* JA’ = J. Particular cases of such A’ are given by A’
=
(i
:), (a = 6,u2 - I b l2 = l), (
(3.6.3)
and, in fact, the general such A’ may be factorized into matrices of this form. It may be shown that the general solution to the problem of finding such XA B can be built up from the elements we have given. For a boundary problem concerning orthogonal polynomials we combine the first matrices in (3.6.2-3), a typical recurrence formula being represented by the transformation
+
(3.6.4)
If we set up a sequence of recurrence formulas of this type and write u,, v, for the entries in the vector yn , this recurrence formula may be written explicitly as un+l = hanun
+ bnvn ,
Vn+l = A&nun
+ anvn
(3.6.5)
Here we assume a, real and positive, and an2- Ib, l 2 = 1 ; the latter may be relaxed to un2 - I b, l2 > 0 at the cost of slight modifications in the formulas.
3.6.
THE
2-BY-2 UNIT
93
CIRCLE CASE
We take the initial conditions uo = 1,
wo =
1,
(3.6.6)
whence (3.6.5) defines u, , v, as polynomials in A, of which the former have orthogonality properties, as we show in Chapter 7. We impose a terminal boundary condition u, = eiuvm, (3.6.7) and the eigenvalues will be the zeros of a polynomial. I n terms of the general formalism (3.1.4-5) for boundary conditions, this corresponds
(3.6.8) We have here M* JM = N * J N = 0, and again M , N have no common null-vectors. Naturally, there are many other possible choices of boundary conditions for example, the periodic boundary conditions given by M=N=E. For the second unit circle possibility we take J = E, the unit matrix. Here we seek 2-by-2 matrices A, B such that h A B is unitary, in the ordinary sense whenever h is on the unit circle. Matrices of this form are again given by (3.6.2), and these solutions may again be extended by multiplying by a general 2-by-2 unitary matrix A‘. We are now confined to “two-point” boundary conditions. Denoting by y o , y m the initial and final vectors of the recurrence sequence, we may impose the condition yn4= Nyo where N is any fixed unitary matrix. Some degree of unification is possible if we extend our consideration from matrices h A B to the bilinear form (XA B ) (hC I))-’. For example, if in the latter expression we make the fractional linear transformation h = (ah’ b)/(ch’ d ) , we derive again a bilinear matrix expression, and any invariance property the original expression had on some A-curve will be translated into an invariance property of the new expression on a A’-curve. There thus ceases to be any basic distinction between invariance on the real axis and on the unit circle. In particular, the investigations of Chapters 1 and 2 may then be included in the cases just discussed of two-dimensional invaria.nce on the unit circle. Naturally, bilinear transformations are not the only ones effecting a mapping between the unit circle and the real axis. Another such mapping is A‘ = 4 ( A l / h ) , which has important applications to the connection between polynomials orthogonal on the unit circle and on the real axis, and many others can be constructed.
+
+
+
+
+
+
+
94
3.
DISCRETE LINEAR PROBLEMS
3.7. The Boundary Problem on the Real Axis Here we summarize briefly the constructions associated with the boundary problem (3.1.1), (3.1.5) for the case that the invariance (3.1.2) holds for all real A. Since we discuss some special cases in detail in Chapters 4-6, and since similar, indeed more general, investigations are given in Chapter 9 we do little more than give the definitions. There will, of course, be parallels with the situation of Chapter 1 as well. We assume that J is nonsingular and skew-Hermitean, and that A,, , B, , n = 0, ..., m - 1, satisfy the conditions noted in Section 3.3 as equivalent to the invariance (3.1.2), namely Bn*JBn = J ,
+ Bn*JAn = 0,
An*JBn
An*]An = 0. (3.7.1)
The boundary matrices M, N are again assumed to satisfy (3.1.4), and to have no common null-vectors. The fundamental solution of (3.1.1) will be a square matrix Y,(A), defined by Yn+l(h)= ( U n Bn) Yn(h), YdX) = E, (3.7.2)
+
so that Yn(x) = (
+ Bn-1)
U - 1
(XAo
+ Bo)*
(3.7.3)
For real A, p we have
C+I(P) JYn+l(x)- Yn*(p)JYn(h) = =
Yn*(p){(tLAn*
+ Bn*) I(% + Bn) - 1)Yn(X) (3.7.4)
- P ) Yn*(t.)Bn*JAnYn(h),
using (3.7.1). Introducing the notation %*]An
=
(3.7.5)
Cn 9
and summing (3.7.4) over n, we get Yrn*(~) JYm(X) - J
3
m-1
=
- P)
Yn*(p)CnYn(h).
(3.7.6)
This forms an analogue of (1.4.1), and again of the Christoffel-Darboux identity for orthogonal polynomials, and of the Lagrange identity for linear differential equations. I n particular, if A, p are real and equal, the right of (3.7.6) vanishes, expressing the fact that Y,,(A) is J-unitary for real A.
3.7.
THE BOUNDARY PROBLEM O N THE REAL AXIS
95
T h e recurrence relation (3.1.1) having the general solution = Yn(A)yo, the boundary problem is equivalent to that of finding h such that
yn
NV = y m = Ym(h).Yo = Ym(h) Mv,
with some v # 0. T h e eigenvalues are thus the roots of det ( N - Ym(h)M ) = 0.
(3.7.7)
With each root A,, of this equation there will be a sequenceof column matrices
+
Yo7
1
...,Y m r
9
such that yn+l,t = Bn)ynT 7 Y* = Mvr 9 ywc, = Nvr 7 where v?.# 0. If A,, A, are two real and distinct eigenvalues, then on multiplying (3.7.6) on the left by y&, and on the right by yor the left-hand side vanishes by the boundary conditions, and we obtain the orthogonality of the eigenfunctions in the form (3.7.8) We now make the further assumption that the C, have constant sign, say, Cn>O,
n=O
,..., m - 1 ,
(3.7.9)
in the sense that the C, are positive semidefinite; they cannot be definite in view of the last of (3.7.1). Furthermore, we make the definiteness assumption concerning the recurrence relation that for any nontrivial solution of (3.1.1) we must have m-1
(3.7.10)
This enables us firstly to ensure that the eigenvalues are in fact all real; by (3.7.10) no eigenfunction can be orthogonal to itself, and it may be shown, in a similar manner to the proof of (3.7.8), that an eigenfunction corresponding to a complex eigenvalue would have just this property. In the second place, the eigenfunctions can be normalized by multiplication by suitable scalar factors, so as to ensure that (3.7.1 1)
96
3.
DISCRETE LINEAR PROBLEMS
If this be done, and we write u, for the initial value Mv, = y& of the rth eigenfunction, the spectral function .(A) may be defined as a stepfunction, whose value for any real A is a square matrix of order K, whose jumps occur at the eigenvalues A,, and are of amount u,u,* . I n a similar manner to Section 1.6, we may define a characteristic function, whose poles are at the eigenvalues and whose residues there are the jumps of the spectral function. A suitable form turns out to be FIM.Jh) =
a (Y;'(A) N + M ) (Y;'(A) N
-
M)-']*-'.
(3.7.12)
A similar function is investigated in connection with the general firstorder system of differential equations in Chapter 9. T o identify the function with that of Section 1.6, we take J = --i, M = 1, N = exp (ia).
3.8. The Boundary Problem on the Unit Circle Here too we confine ourselves to a brief discussion. We take J to be Hermitean and nonsingular, but not necessarily definite. T h e A, , B, in (3.1.1) are now to satisfy (3.4.2-3), so that AA, B, is J-unitary when 1 h I = 1. ,Defining the fundamental solution Y,(A) as previously, it may be shown, for example, by induction, that
+
I n particular, if I A 1 = 1 we have that Y,,,(A)is J-unitary. Writing C , = A,* JA,fi, we assume that C,, 3 0; it is again not possible that C , > 0, except in the comparatively trivial case in which B, = 0. Again we make the definiteness assumption that (3.7.10) holds for any nontrivial solution of (3.1.1). T h e eigenvalues are again the roots of (3.7.7), and by means of (3.8.1) we may prove first that all eigenvalues lie on the unit circle, and secondly that the eigenfunctions are orthogonal according to (3.7.8). We may suppose them normalized according to (3.7.1 1). For the spectral function we consider a weight distribution on the unit circle. With the eigenvalue A,, on the unit circle, we associate the (matrix) weight u,u,*, where, as before, u, is the initial value yor of the corresponding normalized eigenfunction.
CHAPTER 4
Finite Orthogonal Polynomials
4.1. The Recurrence Relation We take up here boundary problems of Sturm-Liouville type associated with the recurrence formula CnYn+i
= (anh
+
bn)Yn
- Cn-iYn-1,
71
= 0,
..., m
- 1,
(4.1.1)
where the a,, b, and c, are real scalars, subject to an
> 0,
cn
> 0.
(4.1.2)
A boundary problem is given if we ask for sequences y-l , ..., ynrconnected by this relation, not all zero, and satisfying the boundary conditions (4.1.3)
where h is some fixed real number. That this is a problem of eigenvalue type, soluble only for isolated values of A, is easily seen if we construct a typical solution, that is to say, sequence, satisfying (4.1.1) and the first of the boundary conditions (4.1.3), and not vanishing throughout. We must, of course, take y o # 0, since otherwise by (4.1.1) y1 = 0, y z = 0, ..., and the sequence vanishes identically. It will be convenient to define a standard solution Y - ~ ( A ) ,~ o ( h )~ ,1 ( h ) i* * * ,
ym(X)
(4.1.4)
of (4.1.1) with the fixed initial conditions y-,(h)
= 0,
yo@) =
l / C 1
> 0.
(4.1.5)
Now that we have fixed y-,(h), yo(h),the values of yl(h), y,(h), ..., are to be found successively from (4.1.1). For n 2 0, it is evident that y,,(A) is a polynomial of degree precisely n. We can now say that the remaining boundary condition in (4.1.3) will be satisfied if (4.1.6)
4.
98
FINITE ORTHOGONAL POLYNOMIALS
The roots of this equation, the eigenvalues, are thus the zeros of a polynomial of degree m. For if (4.1.6) holds, the sequence (4.1.4) certainly satisfies the conditions (4.1.11, (4.1.3) of the boundary problem, without vanishing identically; conversely, it is easy to prove that any solution of (4.1.1) and (4.1.3), not vanishing identically, must be a sequence proportional to (4.1.4) for such a A-value. In showing that the eigenvalues of our boundary problem are the zeros of certain polynomials we begin to approach the theory of orthogonal polynomials. It is not immediately apparent that the polynomials (4.1.4) defined by (4.1.1) and (4.1.5) have any orthogonality properties. This may be deduced from the orthogonality of the eigenfunctions by arguments similar to those of Section 1.4 (cf. Theorem 1.4.5). As in Chapter 1, we are here considering only the orthogonality of finite sets (4.1.4), leaving the infinite discrete case to the next chapter. T h e converse step, of showing that polynomials known to be orthogonal satisfy a recurrence relation of the type of (4.1.1), will be considered later in this chapter.
4.2. Lagrange-Type Identities We collect here for later use some results of the type of Green’s theorem or the Lagrange identity for differential equations, here associated with the names of Christoffel and Darboux. T h e results are analogous to Theorems 1.4.1-2; a more general result was indicated in (3.7.6). They may be used to establish the reality of the spectrum, the orthogonality of the eigenfunctions, and for oscillatory investigations. We have first: Theorem 4.2.1.
For 0
< n < m,
4.2.
LAGRANGE-TYPE IDENTITIES
99
Putting n = 0 and recalling that ~ - ~ ( = h )~ - ~ ( = p 0, ) we derive (4.2.1) with n = 0. Induction over n then yields (4.2.1) from (4.2.2) in the general case. We deduce two important special cases. Dividing (4.2.1) by (A - p) and making p -+ h for fixed h we get, using 1’Hopital’s rule:
Theorem 4.2.2. For 0
< n < m, (4.2.3)
I n particular, for real A,
A+l(4Y n ( 4 - Y?a+l(4 Y X 4 > 0.
(4.2.4)
T h e other special case of Theorem 4.2.1 is
Theorem 4.2.3.
For 0
< n < m, and complex A,
This results immediately on putting p = X in (4.2.1). Further results of this type relate to two distinct solutions of (4.1.1). As a second standard solution we take a sequence Z-l(A),
such that
c,,zn+1(4 = (%A
+
...,%n(A), bn)
(4.2.6)
zn(4 - c,,-lzn-l(4
(4.2.7)
z-,(A)
(4.2.8)
and with the fixed initial conditions z,(h) = 0,
>
=
1.
For n 1, .=(A) will be determined recursively from (4.2.7) as a polynomial of degree n - 1. I n analogy to (4.2.1) we have then:
Theorem 4.2.4. For 0
In particular, for h
= p,
< n < m,
100
4.
FINITE ORTHOGONAL POLYNOMIALS
For the proof of (4.2.9) we take the results
(4 + bn) Y n ( 4 - Cn-1Yn-1(4,
CnYn+1(4
=
CnZn+1(P)
= (%P
+ bn)
Zn(P.> - Cn-l%-l(Ph
multiply, respectively, by z,(p), y,(h) and subtract, getting Cn{Yn+1(4 Zn(P>
- P)Y n ( 4 z n w
- Zn+1(P) Y n ( 4 ) =
+
cn-l{Yn(h) Z n - l ( r >
(4.2.1 1)
- Z n ( P ) Yn-1(4>.
Putting n = 0 and recalling that y-,(h) = 0, zPl(p) = 1, ~ - ~ y ~=( h1,) we get (4.2.9) with n = 0. T h e general case then follows as before by induction over n, using (4.2.1 1). T h e case h = p, (4.2.10), constitutes an analog of the constancy of the Wronskian determinant for two solutions of a differential equation of the form y" u(x)y = 0.
+
4.3. Oscillatory Properties We prove here results concerning the reality and separation properties of zeros of the y,(h), and more generally of polynomials of the form y,(h) hy,-,(h). These results will of course also give information on the spectra of boundary problems of the form (4.1.1), (4.1.3). For the classical polynomials, such as those of Legendre, numerous methods are available for proving such results ;for general orthogonal polynomials, still other methods are available, based on the orthogonality. Here we confine ourselves to methods based on the recurrence relation, and its immediate consequences as found in Section 4.2. A basic result is:
+
Theorem 4.3.1.
For real h, the polynomial Yn(4
+ hz-l(4
(4.3.1)
has precisely n real and simple zeros. Suppose if possible that A is a complex zero of (4.3.1). Using this fact and taking also complex conjugates we have Yn(4
+ hYn-1(4 = 0,
YJJ)
+
hYn-l@)
= 0.
(4.3.2)
We then have that the right of (4.2.5) vanishes, and this is impossible since the terms on the left of (4.2.5) are nonnegative, that for Y = 0 being positive, by (4.1.2), (4.1.5). Hence the zeros of (4.3.1) are all real.
4.3.
101
OSCILLATORY PROPERTIES
That they are all simple follows from the fact that at a hypothetical multiple zero, necessarily real, we should have simultaneously yn(h)
+ hyn-,(A)
= 0,
yn’(h)
+ hyL-,(h)
= 0,
and so y,(h)y~_,(h) - y,’(A)y,-,(X) = 0, in contradiction to (4.2.4). Since (4.3.1), as a polynomial of degree exactly n, must have n zeros altogether, this completes the proof. I n other words, the boundary problem (4.1.1), (4.1.3) has a purely real spectrum, consisting of m real eigenvalues. Next we give some separation theorems. A simple case is
Theorem 4.3.2. Two consecutive polynomials y,(h), y,-,(X) have no common zeros. Between any zeros of one of them lies a zero of the other. Since all zeros are necessarily real, the first statement follows from (4.2.4). The rest of the theorem also follows from (4.2.4). Suppose that A , , A, are two zeros of y,(X), which we take to be consecutive; since y,(X) has only simple zeros, this implies that yn’(hl),yn’(h2)have opposite signs. By (4.2.4), with n - 1 replacing n, we have Yn‘(4) Y n - d U
> 09
b‘(UYn-l(h2) > 0,
and so y,-,(X,), y,-,(A2) must also have opposite signs, which proves the result. The proof that between two zeros of y,-,(X) lies a zero of yn(h) is similar. Generalizing this type of argument, we have the following analog of Theorem 1.3.6.
Theorem 4.3.3. For real distinct h, , h, , between any two zeros of Y A 4 h,Y,-,(h) lies a zero OfY,(4 h,y,-,(X). This may be proved in a similar manner, using (4.2.4). T o put the argument differently, we note that (4.2.4) implies that y,-l(X)/y,(h) is a strictly decreasing function of X when it is finite. As X increases from --oo to +m, y,-,(A)/y,(X) will start at 0, and tend to -00, as X approaches as X goes from the the lowest zero of y,(X), then go from $00 to lowest to the next lowest zero of y,,(h), and so on, finally tending to zero as h 00. Hence between any two A-values at which y?-,(A)/y,(h) = - l/h, , taking it that h, # 0, this function will have a discontinuity, tending in between to --oo and again going from $00 to - l / h l : and hence taking all other values, including -l/h,. T h e proof is similar if h, = 0. The consideration of infinities may be avaided by considering in place of y,-,(X)/y,(A) the variation of the function
+
+
--03
---f
+
b n ( 4 + &I-l(WYn(4 - k l ( 4 1 ,
(4.3.3)
102
4.
FINITE ORTHOGONAL POLYNOMIALS
It follows from (4.2.4) that as A varies on the real axis, this function moves monotonically on the unit circle. We shall use this device in connection with matrix systems. We turn to a different type of oscillation theorem, in which A is fixed, or is treated as a parameter, and yn(A) is viewed as a function of n. So far y,(A) has only been defined for integral values of n, n = -1, 0, ..., m. We complete it to a continuous function y,(A), -1 x m, by specifying that between two integers, n x n 1, y,(A) is to be a linear function of x. This definition may seen artificial, particularly when applied to a classical polynomial such as that of Legendre, for which another and more analytic definition is available for nonintegral orders. T h e definition is however entirely natural from the point of view of the mechanical problem which gives rise to (4.1.1), namely, the problem of the vibrations of a stretched string bearing particles. Here the segments of the string between consecutive particles are, of course, straight, and are appropriately represented by linear functions. We start by observing that the zeros of y,(A), -1 x m, for fixed real A, are simple and well defined. Suppose first that yl,(h) = 0 for some non integral x‘. Then (a/ax)y,(A) exists at x = x’, and is not zero, for if it were, then y,(h) being linear would vanish throughout the interval of the form n x n 1 containing x’, and this is impossible since y,(A), Y,+~(A) cannot both vanish, by Theorem 4.3.2, or by the recurrence relation. Suppose again that yJA) = 0 for some integer n. I t then follows from (4.1.1) that ynpl(A),Y,+~(A) have opposite signs. Hence y,(A) - ~ , ~ - ~ (Y,+~(A) h ) , - y,(A) have the same sign and n is a simple zero, in the sense that y,(A) changes sign as x increases through n, the derivative (a/ax)y,(A) having the same sign and not being zero in (n - 1, n), (n, n I). For any fixed A, y,(A) will have a certain number of discrete zeros in -1 x m, including a fixed one at x = - 1. We can now consider the behavior of these zeros as h varies. It is easily seen that y,(h) is a continuous function of x and A. We have seen that the zeros of y,(A) are simple, corresponding to non-zero values of the derivative (a/ax)y,(A)if x is nonintegral, and to non-zero left and right derivatives, y,(A) - ynWl(A), Y ~ + ~ ( A )- y,(A), respectively, if x is an integer n. From this we deduce that as A varies, the zeros of yz(A) in -1 < x < m vary continuously, as functions of A. T o take this up in detail, suppose that y,.(A’) = 0, - 1 < x’ < m, x’ being not an integer. Then for some E > O the interval x’ - E < x < x’ E will contain no integer, so that y,(A’) will be linear in x in this interval, and y,#-,(A’), ~ , ~ + ~ ( hwill ’ ) have opposite signs. By continuity, there will be a 6 > 0 such that these statements are also true of y,(h)
< < +
<
0 such that if I A - A‘ I < 6, then y,(h) has a unique zero within a distance E of every zero x‘ of yZ(hl)in - 1 < x‘ < m, and possibly also a zero in m - E x m, ify,(X’) = 0. In addition, if B is small enough, y,(h) will have no other zeros in -1 < x m ; this follows from the fact that y,(h) is continuous in both variables, its zeros, for fixed h and varying x, being points of change of sign. We can now show that these zeros are monotonic functions of A.
+
+
A T , and likewise no zeros for X > A,, rn zeros for A < A,,. Moreover, zeros of y,(h) in -1 < x < m are in unique correspondence with changes of sign in the sequence (4.3.5). This completes the proof. It will be observed that the order of the A,. as real numbers is in this case opposite to their order in the oscillatory characterization. This results from the choice of sign of a, in (4,1.1), which ensures that the y,(X) have positive highest coefficient, in keeping with the standard practice in the theory of orthogonal polynomials.
0, then Imf(X) < 0, and conversely. In Section 1.6 we noted some interpolation properties of the characteristic function for that case. T h e place of those is here taken by an asymptotic expansion ~ t f ~ , , ~ (for X )large A, again connected with moment problems. We turn to this aspect later.
4.6. The First Inverse Spectral Problem As in Section 1.7, we consider the reconstruction of the boundary problem given the spectral function; that is to say, we are given the real quantity h, the eigenvalues or zeros of ym(A) hy,-,(h),. and the normalization constants p,. defined in (4.4.3),or (4.4.4).It is easy to
+
108
4.
FINITE ORTHOGONAL POLYNOMIALS
show that the polynomials y,(h), ...,ym(X)are fixed, apart from constant factors, so that the recurrence relation (4.1.1) is essentially determinate apart from certain trivial transformations. As in Section 1.7, a knowledge of the eigenvalues only is insufficient. The solution of the problem may be divided into two stages, each of which is well known in the theory of orthogonal polynomials. First we construct the y,,(h).
Theorem 4.6.1. Let ~ , , ~ ( hbe ) a nondecreasing step function with precisely m points of increase, and let a,, ..., a,,-, be given positive quantities. Then there exist unique polynomials yo@), ..., yT2L-l(h) satisfying (4.5.3), and such that yn(X) is of degree n, the coefficient of An being positive. T h e proof is by “orthogonalization.” T o sketch the process, we seek y,(h) in the form (4.6.1)
and consider first the solution of (4.5.3) with p # q. For these it is necessary and su@cient that m -m
y,(h) h‘ dT,,h(h)
= 0,
q = 0,
...,p
- 1.
(4.6.2)
Substituting (4.6.1) on the left, and introducing the “moments” ui =
J
co
~ldT,,,(h), -53
j = 0,1,
... ,
(4.6.3)
we may replace (4.6.2) by the system of linear equations (4.6.4) r=O
+ +z,r(h)
will cross the real axis in the sense of positive motion around the origin, and its argument will increase through a multiple of 7 ~ .
6.10.
THE MULTI-PARAMETER OSCILLATION THEOREM
Suppose again that y,,,(X) = 0 for some integral n, 0 T h e relevant section of the polygon a joins the points
+
un,r(A) = -Cn-l,rYn-l,r(A) %r(4
165
< n < m.
*n-l.r(A)
= -cn-l.rYn-l.r(49
% + l * r ( 4 = cn*rYn+l.7(4 = %.r(% %+1,r(X)
= Gln.rYn+1.r(4
+
iYn+1,r(9
That the second and third of these four points are identical if y,,,(A) = 0 follows from the recurrence relation (6.9.1). Here Y,-~,~(A), Y,+~,,(A) have opposite signs. If say Y,-~,,(A) < 0 < Y,+~,,(A), it is easily seen that a point describing this section of a will cross the real axis from below to above in the right half-plane, a similar discussion applying to the opposite case. Hence as z describes this part of 6,arg z will increase through a multiple of T . T o complete this part of the argument we observe that if as z describes a, arg z reaches a multiple of 7, then this may occur through the points (6.10.3-4) lying on opposite side of the real axis, corresponding to a nonintegral zero of yZ,,(A), or through one of these points lying on the real axis, corresponding to an integral zero of yZ,,(A). I t is not possible for arg z to be a multiple of 7 at a point of the join of w,,,(h), ~ , + ~ , ~ ( h ) , other than the end points, since this would imply y,,,(A) = 0, and also ya,,(A) - Y,-~,,(A) = 0, which is impossible. Hence as z describes the polygon in the order (6.10.5)) a r g z will be a multiple of 7 only at points corresponding to zeros of yz,,(X), and will pass through these multiples of 7 from below to above. Hence if yz,,(A) has qr zeros for which -1 < x < m, and in addition vanishes at x = m, A being an eigenvalue, we must have q+(h) = (qr
+ 1)T,
T
=
1, ...)k.
(6.10.12)
For as z describes (5, arg x must increase through qr successive multiples of T , its initial value being zero, and since it is initially increasing. In addition it must reach a further multiple of T at the end of &, and the arguments just given show that it can only reach this further multiple of 7 from below. T h e problem now assumes the following form. T h e rpr(A), being k functions of the k real variables A,, are to have the values (6.10.12), and we are to show that the equations (6.10.12) have precisely one solution for A,, ..., A,. We consider the transformation
(4>
***)
A,)
-
(Vl(4,
*.*,
?Jk(4),
(6.10.13)
166
6.
MATRIX METHODS FOR POLYNOMIALS
or in abbreviated form,
-
v(4
(6.10.14)
as a transformation rp of euclidean k-space, 8, say, into some part of itself. Our first task is to identify the range rpb, , the set of all possible sets of values of (rpl ..., fpk) for all possible finite real A. We observe first that y b k is certainly a bounded set. As z describes any side of the polygon a , arg z cannot vary by more than T . We therefore have the crude bounds
I vr(h)I
< (2m + 1)x,
Y
=
1, ..., k.
(6.10.15)
Next we consider the boundary of rpgk. Since the mapping (6.10.13-14) has nonvanishing Jacobian, as was proved in connection with the previous theorem, I p 6 k contains with any point rpt, the map of some At, also a neighborhood of rpt, the map of some neighborhood of At, by the implicit function theorem. Thus r p d k is an open set, and points of its boundary will be the limits of sequences of the form rp(X(j)), j = 1, 2, ..., where the sequence h ( j ) has no finite limit-point. We show that this implies that the boundary of Fdk is located in certain planes. Assume then that h(3) = (Xf), ...,A t ) ) , j = 1, 2, ..., is an infinite sequence such that
(6.10.16) and write
the unit vector p ( j ) prescribing the direction from the origin to X(j). Since p ( j ) is bounded, we may by selection of a subsequence arrange pk) that the sequence p ( j )converges to a limit p, say, where p = ( p l , is also a unit vector. Then XCj) = Jljp
+
O(Jlj),
(6.10.18)
the last term being a k-vector whose entries are small compared to + j as j -+ a. Substituting for X in (6.9.1-2) we obtain., selecting the leading terms,
6.10.
167
THE MULTI-PARAMETER OSCILLATION THEOREM
With the notation (6.10.20) 8-1
this simplifies to
The significance of the latter as an asymptotic formula for Y ~ , ~ ( A ( ~ ) ) for large j will depend on the v ~ not , vanishing, ~ and on their sign. Regarding this we prove that there is at any rate one r, 1 r k, for which the sequence ..., v , , - ~ , ~contains terms only of one sign, and no zeros. Assume if possible that this were not the case, that for each such sequence there was either one zero member or two members of ..., opposite signs. Then for every r we can find a set of scalars none negative and at least one positive, such that
<
0 is given by
T h e modified recurrence relations (7.4.6-7) may be written
and since I b, I < 1, this system satisfies the restrictions laid down in (7.1.4-5). If, in addition, the initial conditions (7.1.3) are to hold, the constant KO is restricted by (7.3.11) in that
1
2n
0
dT(8) = ~ ( 2 7 r) ~(0) = k;’.
182
7.
POLYNOMIALS ORTHOGONAL O N THE UNIT CIRCLE
7.5. Uniqueness of the Spectral Function As in Chapter 5, and earlier in Chapters 1 and 2, we may define a spectral function as one with respect to which the solutions of the recurrence relation are orthogonal, and enquire whether there exists more than one such function. Starting with the recurrence system (7.1.1-5), we term a suitably restricted function T(8), 0 8 277, a spectral function if we have
<
0 since
/
2n 0
min Rlw(B),
gnl(q nu,(^) - g,(q w(e) de. (7.8.18) Q
=
max I w(0) I,
(7.8.19)
w(0) has positive real part and is continuous, and
7.8.
195
ASYMPTOTIC BEHAVIOR
comparing the real part of the left of (7.8.18) with the modulus of the right, we deduce that
by the Cauchy inequality. Hence
J""I h-"u,(X) - g,(X) 0
l2 d6
1; I
< (Q/W)~
g&)
l2 d6.
(7.8.20)
This is essentially the result. Since we may modify (7.8.20) to 0
- g(X) I2 dB < (2 + 2Q2/w2)
I
I'" 0
I gnl(A) l2 d6.
Introducing a factor An, of absolute value unity, on the left and evaluating the integral on the right we have
J""I #,(A) 0
- h*g(h) l2 d6
< 2 4 2 + 2Qz/w2) 2 I a, 12, m
(7.8.21)
n+1
which gives the required result. For an actual approximation to u,(h), and not one in the mean-square sense, we may work from (7.8.20), deriving
Theorem 7.8.2. Under the conditions of Theorem 7.8.1 there holds for I h I = 1 the bound
I u n ( 4 - hngn(4 I For
< n(Q/w)22 I % 12. m
n+l
(7.8.22)
196
7.
POLYNOMIALS ORTHOGONAL O N THE U N I T CIRCLE
by the Cauchy inequality. T h e bound for the sum on the right needed to prove (7.8.22) is immediate from (7.8.20). In both Theorems 7.8.1-2, a connection is manifest between the accuracy of the asymptotic approximation and the smoothness of the weight function.
7.9, Polynomials Orthogonal on a Real Segment T h e factorization (7.8.8) also provides asymptotic expressions for a certain class of ordinary orthogonal polynomials on a finite interval of the real axis. We use the notation of the last section, with the additional assumption that w(0), when considered as of period 27r and defined for all real 8, is an even function. I n other words, we have w(27 - 8) = w(O), ps = p P 8 , and Wl(4
=
M/4,
I I
=
(7.9.1)
1,
in view of (7.8.10). There is still no need to assume w(0) to be realvalued. We write (7.8.8) or (7.8.9) in the form gn(4 W l ( 4 = -gn1(4
W1(;0
+ 4%
(7.9.2)
with the notation (7.8.17). We deduce that, with X = exp (ie),
r =0,1,
..., n -
in view of the fact that h(X) contains no negative powers of the results for f r it follows that
r = 0,
1,
A. Combining
..., n - 1.
Selecting the even part of both sides, that is making the change of variable 0 ---t -8 and combining the results, we have in view of (7.9.1) that
7.9. for the same
=
POLYNOMIALS ORTHOGONAL O N A REAL SEGMENT Y.
By (7.8.17) this is equivalent to
-I': 12
as cos (n
a=n+l
197
e
I
w , ( ~ )cos ye de,
...,
= 0,
-
1.
(7.9.3)
T h e right-hand side being small for large n, we treat this as an approximation to the problem in which the right-hand side is to be zero, that is to say, in which constants a n g ,s = 1, ..., n, are to be determined such that
That they can be so determined, and are unique, is proved as in the discussion of (7.8.6). Subtracting (7.9.3) from (7.9.4), and replacing cos re by cos (n - Y) 8, we obtain
for
Y =
1, ..., n. Multiplying by (&,,
-
CT) and
summing over
Precisely as in the discussion of (7.8.18) we deduce that
Y
we have
198
7.
POLYNOMIALS ORTHOGONAL O N THE UNIT CIRCLE
We have proved the following result, which we formulate first in terms of cosine polynomials.
<
0. We wish to make z --+ 0, and assert that (8.5.8) is also true when x -+ f 0, x -+ 7 - 0, so that the left of (8.5.9) yields zero as E + 0. By hypothesis, we have u, -+ 0, g -+ 0 as x -+ 4 0, and so in order to prove (8.5.8) for x -+ 5 0 it will be sufficient to show that g/u,, is bounded. Since urn([)= 0, g ( e ) = 0 we have
+
+
+
um(f
+
I,
e+r
6)
=
r ( t ) w,(t) dt,
g(f
+
c) =
fyt
Since v,(.$) cannot vanish with urn((),and since v, we have for small e inequalities of the form
so that glum is bounded, and (8.5.8) holds as x --f f
r ( t ) h(t) dt.
, h are continuous,
+ 0. In an entirely
220
8.
STURM-LIOUVILLE THEORY
similar way, the result may be proved for x --+ r ] - 0. Hence making E ---t 0 in (8.5.9) we have
T o complete the proof we observe that this is also true when 6 = a and 7 is the smallest zero of u,,(x) which is greater than a. If the boundary condition at x = a is that u,(a) = 0, that is, if sin a = 0, this has already been proved. If sin a # 0, then glum is finite at x = a, while hum - gv, = 0 at x = a by (8.3.1) and (8.5.3). Hence (8.5.8) is true for x -+ a 0, so that (8.5.10) is available. Similarly, it is available if 7 = b and 6 is the nearest zero of u, to the left of b. We now note that the interval (a, b) comprises a finite number of intervals of the above forms, that is to say, intervals bounded by consecutive zeros of u, or by a zero of u,, and an end-point of (a, b). I n exceptional cases, there may in addition be intervals throughout which u,, vanishes; in terms of the phase variable 8 = B(x, A,) defined in Section 8.4, there will be m 1 intervals in which 8 goes from a to n, from x to 277, and finally from mx to mx 8, and possibly others in which 6 remains a mu1tip)e of n. Intervals of this latter form, in which u, = 0 and so in whichg 3 0, clearly do not contribute to the integrals in (8.5.4). Hence on summing the results (8.5.10) we have
+
+
+
(8.5.1 1)
which is equivalent to (8.5.4), completing the proof of the lemma. Passing to the proof of Theorem 8.5.1, we suppose if possible that w ( x ) as given by (8.5.1) vanishes at all the zeros of u,(x). We apply the result of the lemma, with w in place of g, and w1 in place of h where
We have then w‘ = rwl in view of (8.5.5), while the boundary conditions (8.5.3) hold since they are satisfied by u, , vn . Evaluating for this case the right of (8.5.4), we have
8.5.
AN INTERPOLATORY PROPERTY
22 1
by (8.5.6). Hence the right of (8.5.4) gives
by the orthonormality (8.3.11). In a similar way the left of (8.5.4) becomes b
m-1
a
0
h m j pw2dx = h m z a : .
Hence from (8.5.4) we have
or
Since the A, are in increasing order, this implies that all the a, vanish. This proves Theorem 8.5.1. T h e following interpolatory property follows at once.
Theorem 8.5.3. Let b, , ..., 6 , be any constants, and let xl, ..., x, be zeros of u,(x) which are distinct from each other and from the endpoints a, b, no two such points lying in an interval in which U, = 0. Then there is a unique set of constants a, , ..., am-l such that
2 anun(x,)
m-1 0
=
b, ,
s =
I,
...,m.
(8.5.12)
For if there were not always such a unique set, there would be a set of a , , not all zero, such that
2 anun(x,)
m-1 0
= 0,
s =
1, ...,m.
(8.5.1 3)
Denoting this expression as before by w ( x ) , we should have that w(x) vanished at all the zeros of u,(x). If x = a, or x = b, or both, were zeros of u,, according to the boundary conditions, then these points would also be zeros of w . Any further zeros of u, would not be essentially distinct from 'these, but would lie together with one of the x, , or one
222
8.
STURM-LIOUVILLE THEORY
of a or b, in an interval in which u, = 0 and so in which Y = 0 almost everywhere; however, in such an interval all the un would be constant, and so also w, which accordingly would vanish throughout such an interval. Hence w would vanish at all the zeros of u, contrary to Theorem 8.5.1. The criterion for the zeros x, and the end-points a , b to be distinct in the above sense may be put explicitly as Y l ( 4
< +1)
< ..-< T l ( X r n ) < Y l ( h
(8.5.14)
where as previously rl(x) = J" ~ ( tdt. )
8.6. The Eigenfunction Expansion
The interpolation theorem just proved may be stated in the form that, given any function ~ ( x )a, x b, and any m, we can find a linear combination of uo(x),..., U , - ~ ( X ) which coincides with it at the zeros of u,(x) in a < x < b ; strictly speaking, the zeros should be distinct from each other and from the end-points, and there must of course be at least m 1 eigenvalues. This is already a form of expansion theorem. Furthermore, making m + m and assuming that there are an infinity of eigenvalues, we obtain approximations which are correct at a larger and larger number of points in ( a , b). It was shown by Prufer that there exists a rigorous argument leading from the interpolatory property to the eigenfunction expansion. If the expansion
<
0. If for simplicity we take as boundary conditions
the oscillation theorem will assert that there is an infinity of eigenvalues, all real, and forming an increasing sequence with no finite limit, corresponding eigenfunctions having 0, 1, 2, ... zeros in 0 < < 1. We may
8.7.
SECOND-ORDER EQUATION WITH 'DISCONTINUITIES
227
derive this from Theorems 8.4.5-6 by considering the first-order system for u(x), v ( x ) given by
ul=v,
11'
= v,
0'
=
-[Ap(x)
11'
= 0,
0'
=
-(Ap
v'=
(nl
+ q(x)] + q'")) u,
11,
-[Ap(x-l)+q(~-l)]u,
< x < 61 , El < x < tl + 1, 0
&+1 <x 0, we see that z,(gn) --+ 0 as m --+ 00, so that (8.7.11) is true in the ordinary sense for ( = g,, , mean-square convergence implying ordinary convergence. We may, of course, extend the boundary conditions to y(1) cos /? - y'(1) sin /? = 0.
y(0) cos a - y'(0) sin a = 0,
(8.7.14-15)
More generally we may superimpose a discontinuity of the form (8.7.2) upon the boundary data. Taking (8.7.14) in the form y(0) cot (Y = y'(0) and applying (8.7.2) with n = 0, the effective boundary condition at E=Ois y'(O+) = y(0) {cot a - Ap'O' - p), (8.7.16) where actually the q ( O ) is redundant. Applying similar considerations to the other end of the interval, we are led to the boundary problem formed by (8.7.1-2) together with ~ ' ( 0 )= ~ ( 0(cot )
- Ap'O)),
~ ' ( 1 )= ~ ( 1(Cot ) /?
+ Ap(m+l)),
(8.7.17-18)
where the constants p(O),$cm+l) are non-negative. T h e main effect of these boundary conditions will be that the sums in (8.7.9-lo), (8.7.13) must now be taken over 0, m 1. I n the case q(() = 0, qcn) = 0, we may interpret the equation considered here as that of a string of density $( (), loaded additionally with particles of masses pen). As already explained in Section 0.8, the case in which the string has a particle at each end, a weightless portion of string being fixed to each end so that the system can vibrate, leads to a boundary problem with the eigenvalue parameter appearing in the boundary conditions.
+
8.8.
THE GREEN’S FUNCTION
229
8.8. The Green’s Function
Returning to the general theory of the boundary problem (8.1.2-3), (8.3.1-2), we give the analog, indeed extension, of the results of Section 6.4. We start with the inhomogeneous problem, in which we suppose given a function x ( x ) E L (u ,b), and ask for absolutely continuous functions 97, $ satisfying the differential equations $‘
9J’ = y$v
+ (AP + 4)9J = x,
(8.8.1-2)
and the boundary conditions (8.6.8-9). Provided that A is not an eigenvalue, we show that (8.8.1-2) has a unique solution which is expressed by
so far as ‘p is concerned; this corresponds to (6.4.2). T h e kernel G(x, t , A) is the Green’s function which provides two of the main proofs of the eigenfunction expansion. Here we use it to establish the uniform and absolute convergence of this expansion, under the conditions of Section 8.6. I n addition, it has sign-definite properties which may be used to prove separation theorems, as was done in Section 6.3. Our first task is to establish the existence of Green’s function and to find it explicitly. This we do by solving (8.8.1-2), together with the boundary conditions, by the method of the variation of parameters. In addition to the solutions u ( x ) = u(x, A), and w(x) = w(x, A) of (8.1.2-3) fixed by the initial conditions (8.3.3), we define a second pair of solutions by means of the terminal conditions ut(b) = sin 8,
d ( b ) = cos 8,
(8.8.4)
corresponding to the solution w,(A) of the recurrence relation, defined in (6.1.12-13). Provided that A is not an eigenvalue, ut and w t will not be merely a constant multiple of u and v. Their functional determinant or Wronskian will be written w = w(A) = u(x) d ( x ) - ut(x) .(X) =
u(b) cos 8 - o(b)sin 8.
(8.8.5) (8.8.6)
It follows from the differential equations (8.1.2-3) that the Wronskian appearing in (8.8.5) is independent of x for a x b; putting x = b, we get the left of (8.3.4), whose vanishing determines the eigenvalues. With these notations we can specify the Green’s function.
<
=
5’
a
u ( t ) ~ ( tw-l )
dt. (8.8.14-15)
Substituting in (8.8.8) we get (8.8.3), with G given by (8.8.7).
8.8.
23 1
THE GREEN'S FUNCTION
T o verify the solution we set up the corresponding expression for
#(x). Substituting for s, st in (8.8.9) we obtain
the full expression for
being
).(.I I
b
~(x= ) w-l
2
u t ( t ) ~ ( tdt)
+ ut(x) /' u ( t ) ~ ( tdt) 1 . a
(8.8.17)
It is a routine matter to verify that these satisfy (8.8.1-2) and the boundary conditions (8.6.8-9). The uniqueness of the solution of (8.8.1-2), (8.6.8-9) follows from the fact that if there were two solutions, their difference would satisfy (8.1.2-3) and (8.3.1-2), so that either this difference would vanish or h would be an eigenvalue. The following formal properties of the Green's function are more or less immediate. Theorem 8.8.2. If h is not an eigenvalue, (i) the Green's function is symmetric, that is, G(x, t , 4 = GO, x, 4,
(8.8.18)
(ii) it is continuous in x, t jointly, and absolutely continuous in x, for fixed t, or in t for fixed x; (iii) its partial derivatives have discontinuities when x = t, according to
a
I A) I
- G(x, t , A) ax
?-.t+O
- G(x, t , at
t==z+o
a
a
- - G(x, t , A) ax -
a at G(x, t , A)
I
=~(t),
a
< t < b,
(8.8.19)
= Y(x),
a
< x < 6;
(8.8.20)
2-t-0
I
t-2-0
the pair G, H satisfy in either x or t the differential equations (8.1.2-3) when x f t, and the -boundary conditions (8.3.1-2); (v) for any eigenfunction u,,(x) we have (8.8.22)
232
8. STURM-LIOUVILLE THEORY
(vi) if p is also not an eigenvalue, there holds the resolvent equation G(x, t , A) - G(x, t , P ) = (P - 4
I
b a
G(x,
5, P ) G(5,t , A) p ( 5 ) d5.
(8.8.23)
Of these (i)-(iv) need only a straightforward verification. For (v), we rewrite (8.5.5-6) as u:, = rvn
,
4
+ (Ap + 4)
= (A - h,)pu,
% I
-
Comparing this with (8.8.1-3) we deduce that
I
b
u n ( ~ )=
a
G(x, t , A) (A - h ) p ( t )un(t)4
which is the required result. For (vi) we consider (8.8.1-2) for fixed x and varying A, writing the solutions of (8.8.1-2) and (8.6.8-9) F ~+ a,. We have then
d = .+a
9
+;
+ (+ + 4)pa =
XI
and the latter may be rewritten as
tfii + (PP + q) Fa = (P - X)Pcpa + X. Hence
may be expressed both in the form (8.8.3) and also in the form
j: G(x, 4 PI {(P - 4m v,n(t)+ X ( t ) ) dt =
s”
G(x, 1, P ) X ( t ) dt
+ (P - 4 J
b
G(x, 5, P ) P ( 5 ) d5
s”
(8.8.24) G(5, t, 4 X(t) dt,
on substituting for ~ ~ on ( tthe ) basis of (8.8.3). Since (8.8.3) and (8.8.24) are the same for all continuous functions x, and since G is continuous, there must hold the identity (8.8.23).
8.9. Convergence of the Eigenfunction Expansion We now use the Green’s function to establish the absolute convergence of the eigenfunction expansion under the conditions of Section 8.6; we shall also consider the uniformity of the convergence, in regard to x and in regard to varying boundary conditions. At the center of these investigations is the Fourier expansion of the Green’s function G(x, t, A)
8.9.
233
CONVERGENCE OF EIGENFUNCTION EXPANSION
in a series of the u,(t), taking x fixed. Taking it that h is not an eigenvalue, the Fourier coefficients are given by (8.8.22) as n = 0, 1, ...,
un(x>
h -An'
(8.9.1)
and so we have the formal Fourier expansion, the bilinear formula (8.9.2)
Without considering, in the first place, whether this formula is true, in either the pointwise or the uniform or the mean-square sense, we base ourselves on the Bessel inequality (8.9.3)
J a
I n particular, taking h = i, (8.9.4)
and so (8.9.5)
for some c independent of x ; for it follows from (8.8.7) that G(x, t, i) is bounded, uniformly in x and t, so that the right of (8.9.4) is bounded, uniformly in x. We now strengthen the result of Section 8.6. We have Theorem 8.9.1. Let q(x) satisfy the assumptions of Theorem 8.6.1. Then the eigenfunction expansion (8.6.1) is true, the series on the right being absolutely and uniformly convergent for a x b. We first prove that the series in (8.6.1) is absolutely and uniformly convergent, and then consider whether its sum is equal to ~ ( x ) .For any integers m,m' with 0 < m < m' we have
<
0,
which is impossible by (8.9.10). Thus (8.9.11) holds for all x such that (8.9.12) holds for arbitrarily small E > 0. In view of (8.1.5), an entirely similar argument shows that (8.9.12) holds at x = a and at x = b. Suppose now that (xl, x2) is an interval in which pl(x) is constant, and that it is not contained in any larger such interval. We have therefore (8.9.13)
8.9. that
Q
CONVERGENCE OF EIGENFUNCTION EXPANSION
235
< x1 < x p < b, by (8.1.5), and furthermore that = Vl@l),
&l)
94x2)
= Vl(XZ),
(8.9.14)
since (8.9.12) cannot hold when x = xl, x = x 2 . By (8.1.6-7) we have = q = 0 almost everywhere in (xl,xz), and so the w, are constant in (xl, xz), by (8.5.6) and likewise I/J by (8.6.7). By (8.6.6) we have then
p
d x ) = dXl)
+
$(XI)
42
and from (8.5.5) u,(x) = un(xl)
+ w,(xl)
r ( t ) dt,
x1
Q x d x2 ,
(8.9.15)
21
s2
r ( t ) dt,
21
x1
< x < x2.
(8.9.16)
If J2*r ( t ) dt = 0,that is to say, if r(t) = 0 almost everywhere in (xl , xz), 21 then q~ is constant in (xl, x2), and likewise the u, and so also vl. Hence it follows from (8.9.14) that p)(x) = tpl(x) in (xl, x2). If again Jx2 r ( t ) dt > 0,we have +1
the last series necessarily converging, since ql(x2) is finite. Comparing this with (8.9.15) with x = x2 and using (8.9.14) we deduce that
However, by the argument just used.
0,
Y*(x,A) A(%)Y(x,A) dx u
which is the same as (9.5.17). Hence, if V-l exists, (Y,"V-')*
> 0,
Wl(b,A) (Y,-"V-1)
and so the right of (9.5.14) has the opposite sign to Im A. Turning to the calculation (9.5.14), with the notation (9.5.16) we may write J?M,N(h) ==
so that
8 uv-',
IrnF&fJ@)= (2i)4(+uv-1 - 1 2 V*-'U*) = (4i)-lV*-'(
Now
(9.5.18) (9.5.19)
v*u- U*V) v-1.
v*u- u * v = (M* - N*Y,*-l)J*(M + Y,") - (M*
+ N*Y*-1 b
-
Y,")'
Since J* = -J this reduces to V*U - U*V
and since M * J M
=
= -2(M*
J M - N*Y,*-l]Y;'N),
N * J N to
V*U - U*V
= 2N*Y*-l b =
-2(h
-
(1- ' , * J Y b )
',"
A) N*Y,*-lW,(b,A) Y,",
by (9.5.8) and (9.5.15). Substituting in (9.5.19) we obtain (9.5.14), completing the proof of Theorem 9.5.1. Hence, as stated, F,,,(A) can have only simple poles (cf. Appendix 11), which occur at the singularities of ( M - Y;'N)-l or ( Y , M - N)-'Y,. These are clearly the zeros of det ( Y , M - N ) or roots of: (9.2.9), that 'Is to say, the elgenvalues. We denote the residue of FIM,,J'h)at 'h, b y P, so that near A, there holds the expansion as a Laurent series FM,N(h) =
P,(h - An)-'
+ ...
the omitted terms being regular near A = A,.
(9.5.20)
9.5.
THE CHARACTERISTIC FUNCTION
27 1
In particular, since P,
=
lim (A - A,)FM,N(h),
(9.5.21)
a+,
we have that P, is Hermitean, since the transition (9.5.21) may be made through real A-values. We proceed to evaluate P, as the jump in the spectral function (9.3.26) at A,. Theorem 9.5.2.
The residue of Pn
that of K(x, t, A) being
=
2
a,= 5,
FM,N(A)
z
+an
at A = A, is (9.5.22)
W:9
rz(43w.
(9.5.23)
We denote as before by K, 2 1 the dimension of the set (9.3.8), which is also the number of terms in the sums in (9.5.22-23); if K , > 1, the orthonormalization (9.3.12) or (9.3.17) is supposed to have been carried out. We first show that P, has rank at most K, . For it follows from (9.5.1) that FM,N(A)JY;1( Y,M - N ) = ( M Y;").
+
+
Since the right is regular for all A, substitution of (9.5.20) shows that Pn(A - An)-'JY,-'(Y,M - N )
is bounded in a neighborhood of A,, so that P,]Y-l(b, An) [Y(b,A") M - N ]
(9.5.24)
=0
Of the factors on the right, J and Y-l(b, A,) are nonsingurar, and [Y(b,A,J M - N ] has rank k - K, , in view of our assumption concerning the set (9.3.8). Hence it follows from (9.5.24) that P, has rank at most K, . We complete the determination of P, by considering the singularities of K(x, t, A). By Theorem 9.4.2 the resolvent kernel is regular except at the eigenvalues, namely the roots of (9.2.9). By Theorem 9.5.1 and (9.5.10-12) the singularities of K(x, t, A) are in fact simple poles, and substituting (9.5.20) in these formula we see that the residue is given by K(x, t , A)
=
Y(x,A,) P,Y*(t, A") (A - An)-1
+ ... ,
(9.5.25)
272
9.
GENERAL FIRST-ORDER DIFFERENTIAL SYSTEM
valid in a neighborhood of A,, excluding A, itself. We now use the fact that an eigenfunction yr(x) associated with the eigenvalue A,, that is, for which A, = A,, satisfies the differential equation
Jr:
=
(AA
+ B ) Y , - ( A - A,)
4,
7
together with the boundary conditions. By (9.4.1-3) we have then (9.5.26)
J a
provided that A is not an eigenvalue. Making A+ A, and using (9.5.25) we deduce that ~r(x> =
J
b
Y(X, An) PnY*(t, ~ ( t ) ~ r dt* ( t )
By (9.3.13) this is equivalent to u, = p,
J
b a
Y*(t, A,) A ( t ) Y(t, A,) dt u , ,
or, with the notation (9.5.17), ur = P,Wl(b, A,)u,,
r = n'
+ 1, ...,n' +
K,.
(9.5.27)
Abbreviating W,(b, A,) temporarily to W, , we may write (9.5.27) as
where W:'' is as before the positive definite square root of W, , which is Hermitean. Hence the Hermitean matrix W:"P,W:'' acts as the identity operator on the orthonormal set (9.3.19) of K , column matrices. We proved above that P, was of rank at most K,; since the same conclusion follows for W:1zPnW:/2we see that the latter is of rank exactly K*& , having K, eigenvalues equal to unity, the remainder of its eigenvalues being accordingly zero. Hence W:''P,W:/' is the projector onto the manifold sparined by the set (9.3.19), that is to say,
the summation being over the same set of Y as in (9.3.19). Removing the nonsingular factors W:'' we deduce (9.5.22). We get (9.5.23) on substituting for P, in (9.5.25) and using (9.3.13). This completes the proof of Theorem 9.5.2.
9.6. 9.6.
THE EIGENFUNCTION EXPANSION
273
The Eigenfunction Expansion
We give here a proof, of a more general character than that used in Section 8.6, and one which depends to some extent on principles of complex variable theory. T h e main argument is contained in
<
bo , b > to , and that ~ ( t=) 0 for t > t o . Substituting (9.6.33) in (9.6.34) we derive (9.9.10)
Here we have used the orthonormality of the y n , and have written the integrals as over (a, a),the integrands vanishing over ( t o , 0 0 ) . We have to express the sum in (9.9.10) as a Stieltjes integral, and to simplify this assume that A is not a point of discontinuity of T M . N , b , ( h ) for any R = 1, 2, ..., that is to say, not one of the corresponding eigenvalues, and not a point of discontinuity of T(h); we assume the same concerning --A. This is legitimate since these excluded points form a denumerable set. Since c, as given by (9.3.24) may also be written W
cr =
a
W
yf(x) A(x) p(x) dx = uf
a
Y*(x,AT) A ( x ) p ( x )dx = u,*$(A,), (9.9.11)
the sum in (9.9.10) is
assuming that * A are not discontinuities of in (9.9.10) we get
TM,N,b(X).
Substituting this
292
9.
GENERAL FIRST-ORDER DIFFERENTIAL SYSTEM
for n = 1, 2, ... . Making n -P 00, we may make the limiting transition (9.9.5) in the finite integral in (9.9.12), getting
jr v*Av dx j -
A
-A
$*(A) dT ( A ) $(A)
< k 2j”x*Ax a
(9.9.13)
dx,
and the asserted result (9.9.9) clearly follows on making A
+ 00.
9.10. Limit-Circle Theory
We confine the discussion here to general remarks. Considering the disks (9.8.23) for fixed h with Im h > 0 and as b --t 00, we know that they form for b 2 b, a family of bounded closed sets, each of which is nonempty and includes those for later members of the family, that is, those for greater values of b. We can therefore conclude that the intersection of all of these disks is nonempty; it includes, for example, the limit of the “center” (9.8.11) as b ---t m, or at least a limit-point of the sequence of centers. The situation may be seen more clearly if we consider the limiting behavior of the quantities C, R, , and R, given in (9.8.1 1-13), (9.8.18-20); as may be seen from (9.8.17), C forms the center of the disk, while R;,, R, form together a sort of radius. Using the formulas (9.8.18-20), and also the fact that P -Q
= i-l{Y,*JYb- J } = 2{Im A} = 2{Im A}
1: Y*(t,
A) A ( t ) Y ( t ,A) dt
(9.10.1)
Wl(b,A)
as in (9.5.6-8), (9.5.15), and carrying out slight manipulations in (9.8.1 l), (9.8.13) we have, for the center, C
={E
+ 2(P - Q)-lQ} G = 4J-’ + {(A
-
A) Wl(b,A)}-’,
(9.10.2)
while the “radius” is given in terms of R;l
=
and R,
= {4Q
( P - Q)-lI2
=
[2{Im A} Wl(b,A ) ] - l l 2 ,
+ 4&(P- Q)-1Q}1/2G
(9.10.3)
(9.10.4)
9.1 1.
293
SOLUTIONS OF INTEGRABLE SQUARE
The main point about these formulas is that C, finite manner on the matrix W,(h 4
K1, R, depend
in a
(9.10.5)
= {W,(b,A)>-l.
For b 2 b, , and fixed X with I m X > 0, Wl(b,A) is positive-definite and nondecreasing as a function of b. Hence W,(b, A) will be positivedefinite and nonincreasing as a function of b, and so will tend to a limit as b -+00. Hence C,&l, and R, will tend to limits as b + 00, and hence E, tends to a limit the locus given by (9.8.17), for all @ with @*@ which is also, in some sense, a disk. T h e simplest case is that in which Wl(b,A) tends to a finite limit as b + w, that is to say, in which
O’and writing (9.5.8) in the form Wl(b,A)
= (2 Im A)-l{
W3(b,A) - J/i},
(9.11.9)
we deduce that if the column matrix u is such that u*W3(b,A) u
then u*
< 0,
Wl(b,A) u Q -(2 Im A)-l u * ( ] / i ) u.
(9,ll.10) (9.11.11)
As has been shown, W3(b,A) has k’ negative eigenvalues, and so (9.1 1.10) holds for a linear manifold of column matrices u of dimension k’, namely, linear combinations of the corresponding eigenvectors. Denoting by po any bound from above for the eigenvalues of -J/i,it follows from (9.11.11) that there is a set of u of dimensionality k’ for which u*W1(b,A) u
0, since apart from the trivial solution they are exponentially large as x + m; if I m A < 0, they are exponentially small as x -+ 00 and of integrable square, there being only k” = 1 linearly independent solutions. I n the case of the system (8.1.2-3), in matrix form (9.1.13), we have- J = (! -:), and so k’ = k” = 1 ; as is well known, there is, if I m A # 0 at any rate, one nontrivial solution, of integrable square in the sense that
assuming p , q, Y to satisfy the assumptions of Section 8.1 for all finite b > a. As a final example, consider the system (9.1.18-19), including the fourth-order equation (9.1.16). Here k’ = k” = 2, so that if I m A # 0, (9.1.16) has two linearly independent solutions satisfying
>
here we may assume p , 0, p , > 0, and all coefficients continuous, though these conditions may be much weakened. It was shown in Chapter 5 that if all solutions of a certain recurrence relation were of integrable square for some A, then this was the case for all A. We shall now prove this in the more general case of the differential equation (9.1. l), with an additional assumption. T h e result is
Theorem 9.1 1.2. In addition to the previous assumptions, let J-lA(x) be real. If for some A all solutions are of integrable square, in the sense (9.11.1), then this is so for all A. We assume that all solutions are of integrable square when A = p, and define, for other A, Z(x, A) by Y(X,A)
=
Y(X,P ) Z(x9 A).
(9.1 1.13)
Multiplying on the left by J , differentiating and using (9.1.1) we obtain, writing A, B for A(x) and B(x), (AA
+ B ) Y(x,A)
= (PA
+ B ) Y(&P )
Z(X,
A)
+ /Y(X,P ) q x , A).
9.1 1.
297
SOLUTIONS OF INTEGRABLE SQUARE
Substituting from (9.11.13) on the left and simplifying we have JY(x,p) zyx, A) = (A - p) A Y(X,p ) q x , A).
(9.1 1.14)
or, with the notation (9.11.8), Z ( x , A) = -i(W3(x, p)}-l(A - p) Y*(x,p) AY(x,p) Z(x, A).
(9.11.15)
Abbreviating the latter differential equation to
we now assert that C(x) is absolutely integrable over (a, m), or, in other words,
IrnII 5
C(x)
II dx < m,
(9.1 1.17)
where the norm I I C(x) I I may, for example, be the sum of the absolute values of all the entries in C(x). Since all solutions of (9.1.1) are of integrable square when X = p, we have
in the sense that the diagonal entries of Y*AY, which are non-negative, are absolutely integrable over (a, a),and so also the nondiagonal entries, since Y*AY is Hermitean. Hence to ensure (9.1 1.17) it will be sufficient to show that (W3(x,p)}-l is bounded as x + a. Turning to the proof of the latter statement, we observe first that it is trivial if p is real, since then W3(x,p ) = J/ifor all x. More generally, we have from (9.5.7-8) that
whence, by (9.11.18), W3(x,p) tends to a finite limit as x .+ 03. Thus for its inverse to be bounded as x +m it will be sufficient for its determinant to be bounded from zero as x 3 00. We have, of course, det W3(x,p )
=
det (J/i) det Y(x,p) det Y*(x,p)
298
9.
GENERAL FIRST-ORDER DIFFERENTIAL SYSTEM
and so for the required property, the boundedness of (W3(x,p)}-l, it will be sufficient to prove that
1 det Y(x,p) I > const. > 0.
(9.1 1.20)
By a standard formula from the theory of linear differential equations, we have from the fact that ]Y'(x,p ) = ( p A B ) Y(x,p), Y ( a , p ) = E, the result
+
det Y ( x ,p) = exp
",1
a
t r (pLJ-IA
+ ]-lB) dt1 .
(9.11.21)
Here we note that tr J-lB is purely imaginary, or zero, since tr ]-IB
= tr
(J-'B)*
= tr
B*]*-I
= -tr
I exp
a
BJ-'
=
so that from (9.1 1.21) we have
I det Y(x,p) 1
=
[p
tr (J-lA)dt
-tr LJ-lB,
1I.
(9.11.22)
Since we have required J-lA to be real, so that tr (J-IA) = 0, the right-hand side is unity, proving (9.1 1.20). Returning now 'to (9.1 1.161, having justified (9.1 1.17), we can assert that the solution Z(x, A) tends to a finite limit as x -+0 , and so will be bounded above by some multiple of E . From (9.11.13) we now see that there holds an inequality of the form
1:
Y*(t,A) A ( t ) Y ( t ,A) dt
s:
< const.
Y*(t,p) A ( t ) Y ( t,p) dt,
so that the integral on the left converges as x This completes the proof.
--t 00,
9.12. The Limiting Process u-+
-0,
in view of (9.1 1.18).
b-+
$0
For this purpose, with a view to eigenfunction expansions over the whole real axis, we suppose that a < 0 < b and revise the definitions of the preceding sections so as to replace the value x = a as a basepoint by x = 0. As a fundamental solution of the matrix equation JY' = (AA B ) Y we take the function Y(O)(x,A) satisfying Y(O)(O, A) = E, so that in fact Y(O)(x,A) = Y(x,A) Y-l(O, A). We define a new spectral function T;!,(A) by the properties that it is a nondecreasing Hermitean matrix function whose jumps occur at
+
9.12.
THE LIMITING PROCESS
a --t -00, b -+ $03
299
the A,, and are of amount y,(O)y$(O), thus replacing u, = , ( a ) in (9.3.26) by y,(O). In (9.3.28-29), and in (9.9.8) we are to replace Y(t,A) by Yco)(t,A). The limiting transition to an infinite interval depends on the uniform boundedness of 7'g!N(A), for fixed A, as a -+ --, b -+ Following the limit-circle and limit-point method, we take in this case as the characteristic function K(0, 0, A), where the definition of the resolvent kernel remains unchanged from that given in Section 9.4. Taking x = 0, A = i in (9.7.10) we have
+-.
This gives a bound for the spectral function. T o make it into a bound holding uniformly as a + --, b + we have of course to show that K(0, 0, i> is similarly bounded. Writing F'O)for the new characteristic function K(0,0, A), and dropping the suffixes M, N from the old one, the relationship between them may be written F(0) = Y,FJy-']-'.
+-,
We get this from (9.5.4-5), taking the arithmetic mean of the two according to our definition of K(x, x, A). Writing this in the form F = Y;'F(O)JY,J - l , substituting for F in the equation of the disk (9.8.23), and removing on the left and right factors (JY0J-')*, JY,,J-', respectively, we obtain, assuming as before that Im A > 0,
We now argue that as b increases, or as a decreases, this inequality becomes more stringent, so that the disks "nest," or at least do not expand, and in any case are uniformly bounded, for sufficiently large b > 0, a < 0. This is so since as b increases, the central factor on the right is nondecreasing. In fact, Yay;' = Yc0)(b,A), so that on the right of (9.12.1) we have the factor YfO)*(b, A) (J/i> Y(O)(b,A), which is nondecreasing by (9.5.7). Similarly, Y;' = Y(O)(a,A), and a similar argument shows that Y(O)*(a, A) (I/;)Y(O)(a,A) does not increase as a decreases. Thus the disks (9.12.1) remain bounded. and so also F(O), as a + --, b + +m, and the existence of a limiting spectral function follows from Helly's theorem.
CHAPTER10
Matrix Oscillation Theory
10.1. Introduction
T h e term oscillation refers in the first place to the zeros of real scalar functions, particularly the solutions of second-order differential equations. These zeros have also the meaning that some self-adjoint boundary problem is satisfied, at least in part; this provides a natural basis for interpreting the notion of oscillation in a more general context. Taking the general formulation given by the first-order equation (9.1.1), a new aspect is opened up if we suppress the parameter A, and with a free boundary x1 enquire for what x1 the boundary problem Jy’
= By,
y(.)
= Mw,
y(.J
= NV,
(10.1.1)
admits a nontrivial solution; here as before J is to be skew-Hermitean, B = B ( x ) Hermitean, M* J M = N* J N and M and N are to have no common null-vectors. Such points x1 may be termed “right-conjugate” points of a, relative of course to the boundary conditions; as in the special case of zeros of a scalar solution, we may study separation properties, “disconjugacy” and “nonoscillation.” Reintroducing the parameter, the problem being that of Chapter 9, we have the detailed study of eigenvalues, in particular, separation properties for varying boundary conditions and quantitative information on their distribution. T h e two types of investigation are not quite distinct and may be blended; for example, in Theorems 4.3.4 and 8.4.4 we considered the motion of zeros in x for varying A. Again, there will be other forms of oscillatory investigation, not directly related to boundary problems. In this chapter we discuss these topics first in the context of the vector or matrix Sturm-Liouville system. This will include, of course, the ordinary Sturm-Liouville case of a second-order scalar equation. Treating this as a first-order system as in Chapter 8, and allowing the coefficients to vanish over intervals, we may include also the recurrence relation cases of Chapters 4-5 and Section 6.7. We shall then give the extension to the more general first-order system (10.1.1). 300
10.1.
30 I
INTRODUCTION
T o illustrate the type of question to be discussed we take the trivial case of a scalar first-order equation
-ir’
=[qx)
+ r(x)ly,
a
< x < b,
(10.1.2)
with the boundary condition y(4
=
exp (i.)y(a) f 0,
(10.1.3)
a for-which (10.1.3-4) is soluble may be termed a right-conjugate point of a, with respect to the same boundary conditions; the smallest such b > a may be termed the first right-conjugate point. If (10.1.4) is defined over (a, a), we may raise the question of whether or not there is an infinity of right-conjugate points of a, terming the equation nonoscillatory if there is only a finite number of such points, and oscillatory otherwise. In view of the explicit solution of (10.1.4), namely,
0, and if the equation is disconjugate over (a, b,) for some a, it is also disconjugate over (a, b,) for a’ with a < a’ < 27r; (ii) if Y(X) > 0, all right-conjugate points move to the left as a decreases ; (iii) if Y(X) > 0, then between any two right-conjugate points of a for a = a, lies a right-conjugate point for a = a 2 , a, # a 2 , 0 a,, a2
< 27~;
0, and the equation is nonoscillatory, over (a, a), for some a, it is nonoscillatory for all a ;
302
10.
MATRIX OSCILLATION THEORY
(v) if ~ ( x )> 0, the equation is disconjugate over (a, b) if and only if the least positive eigenvalue of
-+’
= Ar(x)y,
a
< x < b,
with the boundary condition (10.1.3), is greater than 1 ; (vi) between two eigenvalues of (10.1.2-3) for a = a1 there lies an eigenvalue of (10.1.2-3) for any distinct a = az , a1 , az E [0, 27r); (vii) indexing the eigenvalues of (10.1.2-3) in numerical order, there holds the asymptotic formula
as n + +. I n what follows we wish to establish results of a similar character for the general case of the first-order matrix system (lO.l.l), with special reference to Sturm-Liouville systems in matrix terms. An immediate obstacle to this program is that we do not possess an explicit formula extending (10.1 S), for the solutions of (10.1.1) ; the formula Y(X) =
exp
JZ
+ W t)l dt 1 r(4
will be valid only if A(t),B(t) are constant matrices, o r if the system is one-dimensional as just discussed, or if certain commutativity relations hold. T o see how to surmount this difficulty, we may consider an alternative line of argument for the one-dimensional or scalar case (10.1.2-4), not making explicit use of exponential or trigonometric functions. For a solution of (10.1.4) with y ( a ) # 0 we define the function (10.1.7)
where eiu appears simply as a constant derived from the boundary conditions ; this function has affinities with the characteristic function defined in (1.6.1), and so with the notions of a Green’s function or influence function or driving-point admittance. I t turns out that f ( x ) satisfies, independently of a, the Riccati-type differential equation f ’ = -r{f’
+ $1.
(10.1.8)
Without solving this equation, we can draw the conclusion that if
~ ( x )> 0 then f ( x ) is a decreasing function of x. Hence, for example,
10.2.
THE MATRIX STURM-LIOUVILLE EQUATION
303
its zeros alternate with its infinities, that is to say, the right-conjugate points for the boundary conditions y(b) = f e x p (;a)y ( a ) alternate with each other. T o avoid infinities we may consider such expressions as (10.1.9)
although similar transformations may in the present case lead us back to y(x), this will not normally be the case. From the fact that f decreases in x , we can deduce that 8 moves positively round the unit circle as x increases. As B moves from point to point on the unit circle, it will pass through intermediate points, and this observation is a source of separation theorems. By investigating the rate at which 8 moves on the unit circle, estimates for eigenvalues may be obtained. A second obstacle met with in connection with matrix systems is that instead of a scalar quantity moving round the unit circle, we have a matrix moving on the matrix unit circle, that is to say, in the unitary group; we can no longer say that as it goes from point to point, it passes through all intermediate points. Without however looking into the connectivity properties of the unitary group, we may obtain much information by considering the variation of the eigenvalues of the unitary matrix in question.
10.2. The Matrix Sturm-Liouville Equation Suppressing for the moment the parameter A, we consider here the two first-order systems U'
= RV,
V'
=
-QU
V'
=
-Qu,
1
a<x 0, then Z(x, A) is an increasing function of h when A is real, while the eigenvalues of e(x, A) move positively on the unit circle as A increases. For the proof of (10.2.30) we have that
z,= (UV-l),= U,V-' =
UV-lV,V-' u,v-1 - y*-lu*vAv-' -
since Z is Hermitean. Hence 2,
=
v*-1(v* u, - V*V,) v-1,
which is equivalent to (10.2.30) in view of (10.2.29). In the case of (10.2.31-32) we use (10.2.24), replacing U', I/' by U , , I/, and using (10.2.29).
10.3. Separation Theorems for Conjugate Points We compare here the boundary problems in which nontrivial solutions are to be found for U' =
or for
Rv,
U' = Rv,
V'
-Qu,
U(U) = 0,
U(X) = 0
(10.3.1)
-Qu,
U(U) = 0,
V(X) = 0,
(10.3.2)
1
V' =
where Q, R are as before K-by-k Hermitean matrices of functions of x, and u, z, are column matrices. There being no disposable parameter A, these problems will in general be insoluble. T h e x-values, a < x b, for which they are soluble will be denoted by
0, so that y 2 ( x ) > 0, we deduce (10.4.6). Assuming (10.4.7), it follows from the first of (10.4.5) that argw,(b) 3 2 ~ , r
=
1, ..., k,
so that (10.4.8) is soluble for each Y . Thus (10.4.7) is sufficient not only for the existence of C1, but actually for the existence of For further results, which are in some cases more precise than those obtainable by the method just used, we consider the variation of the determinant det O(x), which is of course equal to the product of the w I ( x ) , and use the exact expression (10.2.26). By means of this we prove
ck.
Theorem 10.4.2. Let the assumptions of Theorem 10.4.1 hold, and let 5, exist, a < 5, b. Defining the Hermitean positive-definite or semidefinite matrix I R - Q I by
= sum of absolute values
(10.4.20)
of eigenvalues of R - Q. (10.4.21)
T o verify (10.4.20), write 8 = 8,8,8T, where 8, and 8, are unitary and 8, is chosen so that 8r(R - Q) 8, is in diagonal form; the maximum of tr {(R - Q) 8} remains unaffected if 8 has this form and 8, ranges over all unitary matrices. Now
tr { ( R - Q ) e,e,e,*>
= tr
{ e p - Q) e,e,).
(10.4.22)
Since Q ( R - Q) 8, is diagonal, with as diagonal entries the eigenvalues of R - Q, the right of (10.4.22) is the sum of products of the eigenvalues of R - Q and the corresponding diagonal entries of 8,. Since the latter are of absolute value not exceeding 1, the right of (10.4.22) admits the (precise) bound (10.4.21), which is easily seen to be the same as tr I R - Q I as given by (10.4.9). Hence on taking the trace of (10.4.19) we get and
trQ
< tr(R + Q
+ I R -Q
trQ
3 tr(R + Q
-
I)
(10.4.23)
1 R - Q I).
(10.4.24)
Inserting the upper bound (10.4.23) on the left of (10.4.16) we obtain one of the desired results (10.4.10). For the other, supposing that 5, exists in (a, b] but not Cn+, , we have arg %(b) < 274%
+ 1)
where as before ZinT = n, whence arg det O(b) =
k 1
arg w,(b) < 2 4 n
+ k).
Using (10.4.15) and (10.4.24) as previously we get (10.4.11).
10.5. BOUNDARY
PROBLEMS WITH A PARAMETER
317
10.5. Boundary Problems with a Parameter In this section we consider the matrix Sturm-Liouville equation, reintroducing the scalar parameter, U' =
Rv,
-(hP
V' =
+ Q ) U,
u
< x < b,
(10.5.1)
where P, Q, R are Hermitean k-by-k matrices which are continuous functions of x, and u, v are K-by-1 column matrices. T h e problems to be considered are in a way dual to those of Sections 10.3-4. We now impose boundary conditions in which the independent variable is fixed and A is allowed to vary, the primary cases being those in which we seek nontrivial solutions such that u(a) = u(b) = 0,
(10.5.2)
u(a) = v(b) = 0
(10.5.3)
or such that which are special cases of u(a) = 0,
u(b) cos 4 OL = v(b) sin
+
iy.
(10.5.4)
Once more, we construct a unitary matrix O(x, A ) and consider the behavior of its eigenvalues. Defining U(x, A), V(x,A) by (10.2.27-28), we take O(x, A) = ( V iU)( V - iU)-l and denote its eigenvalues by w,(x, A), r = 1, ..., k. Restricting A to be real, the wr(x, A) will lie on the unit circle, and with their arguments may be taken to be continuous and uniquely defined functions of x and A, subject to
+
arg wr(a,A) argw,(x, A )
= 0,
r
=
1, ..., k,
< argw,(x, A) < ... < arg wk(x,A) < argw,(x, A) + 27.
(10.5.5) (10.5.6)
As shown in Section 10.3, the boundary problems (10.5.2-4) correspond to roots of the equations w,(b, A) = 1, - 1, exp (ia). For separation theorems we rely on a monotonic behavior of the functions arg w,(b, A). Referring to (10.2.31-32), we see that if SZt(b, A) > 0, that is to say, if
J:
U*(t,A) P ( t ) U(t,A) dt
> 0,
(10.5.7)
then (see Appendix V) the arg w,(b, A) are monotonic increasing in A.
318
10.
MATRIX OSCILLATION THEORY
The condition (10.5.7) is of the type of the definiteness conditions (8.2.1) or (9.1.6) ; it may be expressed by the requirement that
i” a
u*(t)
P ( t ) u ( t ) dt > 0
(10.5.8)
for any nontrivial solution of (10.5.1) such that u(a) = 0, since U(x, A) ~ ( a )V(x, , A) .(a) gives the general such solution. T o take a simple case, let us assume that P ( x ) > 0,
a
< x < b,
(10.5.9)
and that R-’(x) exists, for some x in a
< x < b.
(10.5.10)
Supposing if possible that equality holds in (10.5.8), it follows from the continuity of the integrand that u * ( t ) P(t)u ( t ) vanishes identically, and so from (10.5.9) that u ( t ) vanishes identically. From (10.5.1) it then follows that o(t) is constant in [a, b ] . Since u’ also vanishes identically, so does Ro,by (10.5.1) ; by (10.5.10) it follows that o vanishes for some x in [a, b], and so everywhere, since it is constant. Thus the conditions (10.5.9-10) ensure that the functions arg u7(b,A) are monotonic increasing in A. The following separation theorem is now immediate.
Theorem 10.5.1. Let P(x), Q(x), R(x) be continuous and Hermitean, and satisfy (10.5.9-10). Then a closed A-interval containing k 1 eigenvalues of the problem (10.5.1-2) contains an eigenvalue of the problem (10.5.1), (10.5.3) in its interior; the statement remains true with the problems interchanged. Here eigenvalues of these problems are to be counted according to multiplicity, this being the number of linearly independent solutions of (10.5.1-2), or (10.5.1), (10.5.3). Suppose that the interval [A’, A”] contains k 1 eigenvalues of (10.5.1-2). Then, as A increases in this interval, at least one of the u,(b, A) must attain the value 1 more than once, and so at least one of the arg w,(b, A) must increase from one multiple of 27~to the next greater multiple of 27~,and must therefore in between equal an odd multiple of rr. This proves one statement of the theorem, the other being proved similarly. It is a standard circumstance in Sturm-Liouville theory that the
+
+
+
10.5.
319
BOUNDARY PROBLEMS WITH A PARAMETER
eigenvalues can only accumulate at +m. We now give additional conditions which ensure this in the matrix case.
Theorem 10.5.2. Let P(x), Q ( x ) , and R(x) be continuous and Hermitean in [a, b], P(x) and R(x) being also positive-definite. Then the boundary problem (10.5.1), (10.5.4) has at most a finite number of negative eigenvalues. The eigenvalues will be the roots of arg w,(b, A)
=a
+ 2nx,
(10.5.11)
0, then the w,.(x) move positively on the unit circle when at 1. In the notation of this section, we have that for any fixed A, the arg w,(x, A) are increasing functions of x when at a multiple of 277 [cf. Theorem 8.4.3 (ii)]. In view of (10.5.5) we have therefore, for any real A,
+
arg w,(x, A)
> 0,
a
< x d b.
(10.5.12)
Hence in (10.5.11) there will be no solution for n = -1, -2, ... . The assertion of the theorem now follows from the fact that the arg w,(b, A) are monotonic increasing in A. It follows that under the conditions of the last theorem the eigenvalues of the boundary problem (10.5.1), (10.5.4) may be indexed from 0 upwards in ascending order, so that A,
< A, < A, < ... .
(10.5.13)
Under the same conditions we may show that for large n, A, is bounded by multiples of n2.
Theorem 10.5.3. Let P(x), Q ( x ) , and R(x) be continuous and Hermitean in a, b, with P ( x ) > 0, R (x) > 0. Denoting the eigenvalues of the problem (10.5.1), (10.5.4) as in (10.5.13), there is a constant no such that, for n > n o , 2wn < A , ! , ~a z ~ t r { R + P + I R - P l } d x + ( 3 k - l ) ~ , (10.5.14)
27rn >
J”” t r ( R + P - 1 R - P 1) a
where I R - P I denotes d ( R - P)2.
dx - (3K
+ 3)
7r,
(10.5.15)
320
10.
We take it that 0 of the congruence
< 01
MATRIX OSCILLATION THEORY
< 2rr, and denote by n, the number of solutions
arg w,(b, A) = a (mod 27),
A
+
< A,
,
so that Z t n , = n 1, the number of eigenvalues not exceeding A., Since the arg ~ , ( b A), are positive, by (10.5.12), and monotonic increasing in A, we have
arg %(b, A,) 2
a
+ 2(n,
-
1) T,
and so summing over r, arg det B(b, A,)
k
=
arg w,(b, A,)
2 ha
1
Similarly, arg w,(b, A,) < a and so
+
+ 2(n + 1 h,7,
In order to estimate O(b, A) we take it that A auxiliary matrix = B t ( x , A) =
k ) ~ . (10.5-16)
< Ka + 2(n + 1) T.
arg det B(b, A,)
Bt
-
(V
> 0,
(10.5.17)
and set up an
+ iU dA)( V - iU dA)-l.
( 10.5.18)
If we define Ut = Ut(x,A) = d A U ( x , A), then U and V will satisfy the differential equations Ut'
=
dARV,
V'
=
-(dAP
+ Q/dA)U t .
( 10.5.19)
The matrix O t given by (10.5.18) stands in the same relation to the system (10.5.19) as does 8 to the original system (10.2.27). We assert that
I arg det B(x, A)
- arg det Bt(x, A)
I < Kr.
(10.5.20)
T o see this we set up the relation between the eigenvalues of 8 and O t . Let w be an eigenvalue of 8, so that for some column matrix w # 0 we have Ow = w w , or, with z = ( V - iU)-lw, ( V iU)z = w( V - iU)z, or ( w - 1 ) V z = i(w + 1 ) uz. Hence ( w - 1) ( V f i d A U ) z = i{(w 1) f z/h(w - 1)) Uz,
+
+
10.5.
BOUNDARY PROBLEMS WITH A PARAMETER
32 1
and so
This is equivalent to 0twt = wtwt, with (10.5.21)
where w t = ( V - i l / A U ) x. Defining as before the eigenvalues wr of 0, we may form the eigenvalues wS of 0t according to the transformation (10.5.21). We now observe that for fixed positive A the transformation (10.5.21) maps the unit circle in the o-plane continuously into the unit circle in the wt-plane, mapping the upper and lower halves of the unit circle and the points f l into the same entities. We define arg wJ(x, A) initially by argwJ(a, A) = 0, as we may since &(a, A) = E, and thence by continuous variation in x. We can then assert that
since w,(x, A), wJ(x, A) both start at + I when x = a, and remain as x increases within an angular distance m of each other; as we noted in connection with (10.5.21), orand w; must lie either both in the upper half of the unit circle, or both in the lower half, or both at f l or at -1. Hence
and this is the same as (10.5.20), since Z: arg wr(x, A) = arg det 0(x, A), and likewise Zf arg wJ(x, A) = arg det &(x, A) in each case by continuity and since the result is true when x = a. The bounds (10.5.16-17) may then, if n is so large that A, > 0, be put in the form of bounds for arg det &(b, A,). By (10.5.20) and (10.5.16) we have (10.5.22) arg det Ot(b, A,J > ka 2(n 1) T - 3km.
+
+
Similarly, from (10.5.17), arg det Ot(b, A,) < ka
+ 2(n + 1) m + km.
(10.5.23)
We now use, with due alteration, the results (10.4.15) and (10.4.23-24). Since &(x, A) is formed from the system (10.5.19) instead of (10.2.27),
322
10. MATRIX
OSCILLATION THEORY
+
we replace R and Q in (10.4.23-24) by R .\/h and P v ' A Q/dA, respectively. Using (10.4.15), adapted to the present case, we have therefore, for any h > 0,
(10.5.24)
and
Taking it that h
(10.5.25)
> 1, these imply that
and arg det O+(b,A) 2 dA
tr { R a
1 + P - I R - P I} dt + 0 (--) l/h
.
(10.5.27)
Taking for example (10.5.24), it is obvious that the first term Q/dA on the right contributes only O(l/t/A) to the integral; in the case of the second term, it will be sufficient to show that tr I R
-
P -&/A
I - tr I R - P I
=
O(l/A).
This is so since the eigenvalues of R - P - (?/A and those of R - P differ by O( 1/A), and these two traces are the sums of the absolute values of these eigenvalues, as noted in (10.4.21). In particular, we have from (10.5.27) that arg det Bt(b, A ) + 03 as A + m.
(10.5.28)
This depends on the observation that the integrand in (10.5.27) is positive. T o see this let cl, ..., f k be a set of normalized eigenvectors of R - P,so that, by (10.4.21), tr I R
Since R
-
PI
> 0, P > 0, we have
2 I 5:(R - P ) 5, I. k
=
1'
10.6.
323
A FOURTH-ORDER SCALAR EQUATION
and so
as was to be proved. Hence (10.5.28) holds and so, by (10.5.20), as h + m. This shows incidentally that A, exists arg det 6(6, A) 4 for arbitrarily large n > 0; that A, + 00 as n + is evident from ( 10.5.16). Combining now the bounds (10.5.22-23) and (10.5.26) we have, for n so large that A, > 1, ha + 2 ( n
+ l ) w -3kw
/
< +,,
b a
I R -PI}dt
tr{R + P +
1 + 0 (-),A
(10.5.29) ha
+ 2(n + 1) w +
> l/h,,
/
b a
tr { R
+ P - 1 R - P 1) dt + 0 (--)I . d A 7 3
(10.5.30) For large n, these imply the bounds (10.5.14-15) which are the assertion of Theorem 10.5.3, since 0 < 01 27, and the terms O(l/dAn) will be less than 7 in absolute value if n is large.
O, qzo, r > 0 , (10.6.4) and for simplicity that they are continuous in [a, b ] , though Lebesgue integrability would generally suffice; it would also be possible to allow p , or T in the matrix form, to vanish over intervals. T h e conditions (10.6.4) ensure that Q is positive-definite, though not R . T h e form u’ = Rv, v’ = -Qu for (10.6.1) lends itself most naturally to the discussion of such boundary problems as u(a) = u(x) = 0,
(10.6.5)
= v ( x ) = 0;
(10.6.6)
) . ( U
there being no disposable parameter A, there will be at most a discrete set of x-values for which these problems admit a nontrivial solution, the right-conjugate points of a relative to the boundary conditions in question. In terms of the original differential equation (10.6. l ) , these boundary problems are, respectively, y ( a ) = y”(a)
= 0,
y(a) = y”(a) = 0,
y = y”
(10.6.5’)
= 0,
(y”/r)’ = y‘
= 0,
(10.6.6‘)
the second pair of conditions to hold in each case at some x. As before, we denote the solutions, if any, of (10.6.5-6) in (a, b] in ascending order by 51 9 5 2 , and 71 9 q 2 , *.* * The method of Sections 10.2-3 calls in this case for 2-by-2 matrices U , V such that U‘ = RV, V‘ = -QU, U(a) = 0, V ( a ) = E ; we define, when possible, the Hermitean or, in this case, real-symmetric matrix 2 = UV-l, and the unitary matrix 6 = ( V iV)(V - iV)-l. T h e eigenvalues wl(x), w 2 ( x ) of 0 will be such that their arguments are continuous, with arg wl(a) = arg w2(u) = 0, * * a
+
arg wl(x)
< arg w2(x) < arg wl(x) + 277.
Since Q, R are not both positive-definite, we cannot assert that the wr move positively on the unit circle throughout. Since, however, Q > 0, we assert that they move positively when at the point - 1. T h e proof
10.6.
A FOURTH-ORDER SCALAR EQUATION
325
is similar to one given in connection with Theorem 10.3.3. Supposing -1 to be an eigenvalue, and so Ow = - w for some w # 0, we write z = (Y- iU)-lw, w = V z - iUz,sothat ( V iV)z = -(V - iU)z, and so Yz = 0, w = -iUz. Hence, with the notation (10.2.20),
+
W*QW = 22*(V*RV
+ U * Q U ) z = ~ z * U * Q U Z= 2W*Qw > 0.
Hence the w,(x) move positively as x increases when at - 1. T h e following separation theorem is an immediate consequence. Theorem 10.6.1. If for some. n 3 3 there exists qn , then 5n-2 exists and a < -{(M+ B)Y}*Ya+ Y*{AY + (AA + B ) 3 )
=
Y*AY.
(Y*JYa)’ = -(/Y’)*YA
Since Y A= 0 when x
=
a, we deduce that
Y*(x,A) JYA(X,A)
=
J’a Y*(t,A) A ( t ) Y(t,A) dt.
332
10.
MATRIX OSCILLATION THEORY
Substituting in (10.7.14-15) we obtain 52+ =4(V*
+iu *)-1M*r
Y*(t,A)A(t)Y(t,A) df M ( V - iu)-l.
a
(10.7.16)
Since A 3 0, we deduce that SZt
2 0.
(10.7.17)
10.8. Conjugate Point Problems Our approach to the boundary problem (10.7.1-2), where x is to be found, is based on a study of the eigenvalues of the unitary matrix O(x) defined in (10.7.7). As previously, these eigenvalues may be taken to be k continuous functions wl(x), ..., w k ( x ) , their arguments being also continuous and subject to argw,(x)
< ... < argw,(x) < argw,(x) + 2
~ ;
they will be fixed uniquely if we fix their initial values at x to (10.8.1). We have initially
(10.8.1) =a
qU)= { ( M + N ) + ~ J ( M- N ) } { ( M + N ) - ~ J ( M- ~ ) } - 1 ,
subject (10.8.2)
and the initial values of the arg wr(x) will of course depend on M and N ; they will, for example, all start at zero in the case of the periodic boundary conditions M = N = E. We first take up the case B > 0, which is similar to that of (10.3.1) when Q > 0, R > 0, as in Theorem 10.3.1. We show again that the eigenvalues of O(x) move positively, this being the source of separation theorems. Theorem 10.8.1. Let B(x) be positive-definite, Hermitean, and continuous in a x b. Then the functions arg u , ( x ) are strictly increasing in x. It follows from (10.7.11) that 52 3 0 if B > 0, and so under the assumptions of the theorem the arg w,(x) are at any rate nondecreasing. T o prove that they are actually increasing functions, we need the property that if w is an eigenvector of 8, then w*Qw > 0. For this we shall express 52 in terms of 8, B, and J . From (10.7.4-5) we have
<
K solutions, that is, conjugate points, of the problem (10.7.1-2), then it contains at least n - K conjugate points of a according to (10.8 11). For further deductions let us make the simplifying assumption that J*J = E. The rate of change of a simple eigenvalue wr(x) of 8(x) is given by
+
(d/dx) arg w,(x) = w*Qw,
where w is a normalized eigenvector of 8, corresponding to the eigenvalue w,.. Taking w,. = exp (ia), and writing z = '(V - iU)-lw, the form (10.7.1 1) for D gives here (d/dx) arg w,(x) = 4 z*M*Y*BYMa.
Using (10.8.8), and the simplification available from (10.8.13) if J*J = E, we derive (d/dx) arg w,(x) = 4 z*N*(E cos a
+
1-1
sin a)*B(E cos a
+
J-l
sin a)Nz. (10.8.15)
If there is an 01 such that the right-hand side is positive for all z with N z # 0, for example, if z*N*BNz > 0 when N z # 0, it can be asserted that the w,.(x) move positively on the unit circle when at exp (ia);the argument is still valid when w, is a multiple eigenvalue, and (10.8.15) will still be true as regards sign. By this means, separation theorems can be set up for the case when B is not positive definite but satisfies some weaker condition ; essentially this situation was considered in a special case in Theorem 10.3.3.
10.
336
MATRIX OSCILLATION THEORY
As in Section 10.4, the phase differential equation (10.7.10) may be employed in a quantitative sense. Using the form (10.8.3) for 52, we obtain in place of (10.8.15) the result, after slight simplification,
($1
arg w7(x) = w*
;I*
E cos - + 1-1 sin - B (Ecos a
(
2
With the assumption that J * J
=
+ 1-1
“12
sin - w.
(10.8.16)
E, the factor
will be unitary, and the right of (10.8.16) will lie between the greatest and the least of the eigenvalues of B . By this means we obtain bounds for the rate of change of argw,(x), which are valid also when w,(x) is a multiple eigenvalue, and hence bounds for the intervals between conjugate points.
10.9. First-Order Equation with Parameter We indicate here some reasoning parallel to that of Section 10.5, and relating to the eigenvalues of the boundary problem of Chapter 9, namely, Jy’
=
(AA
+B)y,
~ ( 0= )
MU,
y(b) = NU.
(10.9.1)
Once more we consider separation theorems, for eigenvalues for varying boundary conditions, and bounds for eigenvalues or their order of magnitude. As to varying boundary conditions, we may consider (10.9.1) as a particular case of a family of boundary problems with N replaced by N(a), as given by (10.8.10). With the simplifying assumption J*J = E, the problems are given by Jy‘
= (AA
+B)y,
y(a) = M v ,
y(b) = (Ecos a
+
J-l
sin a) NU, (10.9.2)
for any real a. For example, taking M = N = E, and a = 0, T , a pair of boundary conditions which are comparable for our present purpose are the periodic conditions y(a) = y(b) and the antiperiodic conditions y(a) = -y(b), and a separation theorem concerning the associated sets of eigenvalues h will hold under certain conditions. We have
10.9.
337
FIRST-ORDER EQUATION WITH PARAMETER
Theorem 10.9.1. Let the assumptions of Section 9.1-2 hold, and I eigenlet also J * J = E. Then in a closed A-interval containing k values of the problem (10.9.1) there lies at least one eigenvalue of a problem of the form (10.9.2). We define the eigenvalues w,(x, A) of O(x, A), to be fixed at x = a subject to (10.8.1), to be continued thence by continuity and so as to satisfy (10.8.1). Considering the w,(b, A) as functions of A, we have from (10.7.13) and (10.7.16) that the w,(b, A) move positively on the unit circle with increasing, real A. Here we rely on the definiteness condition (9.1.6), showing that the right of (10.7.16) is positive-definite, and not merely semidefinite. By the familiar argument, if in some closed 1 A-interval the wr(b,A) assume the value +1 altogether at least k times, then one of them, at least, must make a complete circuit of the unit circle, and so take all other values on the unit circle, yielding a solution of (10.9.2). More generally, if in this closed A-interval there are n > k eigenvalues of (10.9.1), there are in the interior at least n - k eigenvalues of any other problem (10.9.2). Turning to bounds for the eigenvalues, we may obtain some information in a simple manner from (10.8.16). Replacing B by AA B, where A is real, and we assume that J*J = E, we have
+
+
+
d dx
- arg w ~ xA), = w*
(Y
+
1-1
sin
* (AA + B ) ( E cos + 1-1 (Y
2
7
sin - zo; 2
if w, is a multiple eigenvalue, this holds in the sense that the left lies between the greatest and least possible values of the right for all w with w*w = 1. If we write min (AA B), max (AA B ) for the least and the greatest eigenvalues of AA B it follows that
+ +
min (AA
+B)
0 it follows from the second of (10.9.3) that 2nm
7 < A,
fmax ( A )dx + const. a
with a similar inequality for n < 0; our present assumptions do not ensure the existence of an infinity of eigenvalues of either sign. In the case (10.9.5) we may now assert that A, is of order precisely n. These results are of course sharper than the statement (9.2.4). They become fairly precise in the trivial case in which A is a multiple of E .
CHAPTER1 1
From Differential to Integral Equations
11 .I. The Sturm-Liouville Case In the classical investigation of boundary problems for the scalar second-order differential equation y"
+ (Ap + q ) y
= 0,
a
< x < b,
(1 1.1.1)
we commonly assume that the coefficients p, q are continuous, or at least Lebesgue integrable. Since this form does not cover the case of a second-order difference equation, the topic of Chapters 4-5, we adopted in Chapter 8 the device of extending.( 11.1.1) to a system 10 = yo, o' = -(Ap q) u, the coefficients p , q, and r being piecewise continuous, or at any rate integrable. This procedure still leaves a slight area uncovered, and we outline here another approach, in which we abandon the formalism of the differential equation. Taking one-point boundary conditions
+
y(a) cos
- y'(a) sin a = 0,
y(b) cos j? - y'(b) sin j?
= 0,
(1 1.1.2-3)
we concentrate attention on the solution y(x, A) of (1 1.1.1) such that y(a, A)
= sin a,
y'(a, A) = cos a,
(11.1.4-5)
so that (11.1.2) is automatically satisfied. For this solution we derive an integral equation, of Volterra type, by integrating (1 1.1.1) twice over (a,x) and using (1 1.1.4-5). The first integration gives, using (1 1.1.5), y y x , 4 = cos a -
{Ap(t) 339
+ q ( t ) } y ( tA), dt.
(1 1.1.6)
340
11.
FROM DIFFERENTIAL TO INTEGRAL EQUATIONS
Integrating once more and using (1 1.1.4) we derive y(x, A) = sin a = sin a
+ (x - a) cos a
-
+ (x - a) cos a -
1: s:
s:
{Ap(t)
+ q ( t ) }y(t, A) dt ds (11.1.7)
(x - t ) {Ap(t)
+ q(t)}y(t, A) dt.
Let us now write uo(x) =
r p ( t ) dt, a
(1 1.1.8-9)
ul(x) = s Z p ( t ) dt. a
T h e integral equations (1 1.1.6), (1 1.1.7) may then be written y y x , A) = cos a -
r(t,A) d{Auo(t) + .l(t)>,
( 11.1.10)
and y ( x , A) = sin a
+ (x - a) cos a
-
s:
(x - t )y (t ,A) d{Auo(t)
+ u l ( t ) } . (11.1.11)
T h e differential equation has thus been replaced by an integro-differential, or an integral equation of Volterra type, in which the coefficients of the original differential equation appear only by way of their integrals. We now remark that (1 1.1.I 1) remains intelligible on the basis that y is to be continuous in x, and that uo(x), ul(x) are of bounded variation over a x b. I n some ways this forms the most natural and general framework for problems of Sturm-Liouville type; we mentioned in Section 0.8 the case of the vibrating string, in which ul(x) = 0 and u,,(x) is the mass of the segment (a, x] of the string. Assuming that uo , u1 are also right-continuous, we may derive (1 1.1.10) from ( 1 1.1.11) with the interpretation that y’(x, A) is a right-derivative for a x b, and in fact a full derivative, left and right, if uo and ul are continuous at x. T o verify that formal differentiation of (1 1.1.11) does in fact yield (1 l.l.lO), with due restriction, we use (1 1.1.1 1) for x = x1 , x 2 , subtracting the results and getting, after slight reduction,
<
- 4%- 0)I w(a,)),
11.
344
FROM DIFFERENTIAL TO INTEGRAL EQUATIONS
in verification of (1 1.2.15) with s = 2. T o complete the proof of (1 1.2.15) we use induction. Supposing that (1 1.2.15) is valid for y(aJ on the right of (11.2.1 I) we deduce that
r=1
where we interpret a, = a, ~ ( a = ) 0. Hence
= c3 exp [c4
4411
proving (11.2.15). Inserting this bound on the right of (11.2.7), in the weaker form I y(uJ 1 c3 exp [ c p ( b ) ] , we obtain
0, for otherwise we should also have p = 0, whereas p and q have no common zeros. Hence, for Im A > 0 S(x, A) is regular and satisfies I S(x, A) I 1. Similarly, if I m A < 0, S(x, A) is regular except for poles, has no zeros, and satisfies I S(x, A) I 2 1. T o investigate whether or not S(x, A) is constant as A varies we consider its derivative at A = 0. Indicating a/aA by a suffix A we have, of course,
0. If now for some such h we had 1 S(x, A) 1 = 1, it would follow from the maximum modulus principle that S(x, A) is constant, contrary to (12.2.10). A similar argument shows that if Im h < 0, then 1 S(x, A) I > 1. I n the course of the proof of the last theorem we located the zeros and poles of S(x, A), in relation to the sign of I m A.
+
+
0, and q(x, A) only if I m A < 0, in view of (12.2.15) and the fact that p and q cannot vanish together. That the zeros and poles of S(x, A) are located at complex conjugate points follows from the fact that I S(x, A) I = 1 when A is real, together with the Schwarz reflection principle. More directly, we see from (12.2.5-6) that (12.2.20) so that the zeros of p and q are located at complex conjugate points, proving the result. As already indicated, the condition (12.2.10) for the S-function not to be a constant has a parallel in such conditions as (8.2.1) or (9.1.6), that a nontrivial solution of the differential equation should not be of zero mean-square. I n the present case we prove
Theorem 12.2.3. If the assumptionsof Theorem 12.2. I hold, regarding
a ( x ) , and if, for some x
> 0,
(12.2.21)
holds for one real A, then it holds for all real A. For (12.2.21) implies that aS(x, A)/aA = 0, by (12.2.17), for the real A in question. As we showed in the special case A = 0, this is incompatible with the property that I S(x, A) I 1 for Im A > 0, unless S(x, A) is a constant. I n the latter event aS(x, A)/aA = 0 for all real A, so that (12.2.21) follows from (12.2.17).
0 and that p ) ( a , A) is continuous for real A, with ~ ( 0 0 0) , = 0. T h e question of the location of the branch points of
12.3.
A NON-SELF-ADJOINT
377
PROBLEM
r(m, A) and
~ ( m A), , for the standard case of a differential equation, was raised by Bellman and discussed by Fort and by Levinson and Kemp, in the latter case by similar analysis to that of Sections 12.1-2. T h e third interpretation of the boundary problems associated with the zeros and poles of S is most simply expressed in the case x = 03, and corresponds more closely with the notion of “scattering.” It is an easy consequence of Theorem 12.1.1 that the integro-differential equation
[z’]
+ 1z{dt + A du} = 0,
or the integral equation z(x) = z(O)
+ xz’(0) -
Iz
(12.3.13)
(x - t ) z ( t ){dt
0
+ A do(t)},
(12.3.14)
has a pair of solutions of the asymptotic form, for fixed A, real or complex, and large positive x, zl(x) = eiz
+ o(l),
zz(x) = e-lz
+ o(1).
(12.3.15-16)
T h e problem is now posed of finding S such that the solution of (12.3.14) given by ( 12.3.17) z = sz, - z 2 , where S is independent of x, satisfies the initial condition z(x) = 0. O n comparison with (12.2.7) and recalling that y(x, A) is a solution of (12.3.14) with y(0, A) = 0, we have S = S(m, A), provided that the latter is finite. Thus the poles of S(m, A) are the A-values for which the determination of S as above is impossible. We now show that S(x, A ) does indeed have at least one zero and pole, provided that it is not a constant.
<
O,
kfO.
(1 2.5.16)
We rely in this case on a transformation of the integral equation y ( x ) = y(0) cos k x
+ y'(0)k-'
sin kx - k-l
J: sin k(x
- t ) y ( t )du(t).
(12.5.17)
It may be shown by means of Theorem 1 I .4. I that this is equivalent to (12.5.2); the equivalence is well-known in the case when ~ ( x is) differentiable and in which we have to deal with differential equations. Here we shall establish a modified form of (12.5.17) by a different argument. Defining z(x) = eikzy(x), (12.5.18) ,
-
and substituting in (12.5.17) we get, after slight reduction,
+ I ) + ~'(0) (2ki)-l(eZki~- 1 )
z(x) = +y(O) ( P i '
42kil-l
j' (exp [ 2 ~ i ( x- t ) ] 0
-
1 ) z ( t )do(t).
(12.5.19)
12.5.
GENERALIZATION OF
y"
+ [k2+ g(x)] y
=
387
0
T o derive this directly from (12.5.1 1-12) we have, on taking a
=
0 there,
Multiplying the first by exp (2kix) and subtracting the second we get
On dividing by 2ki we obtain (12.5.19), noting that (12.5.20)
z ( x ) = (2ki)-l[p(x) eZki2 - q(x)l9
+
and also that p ( 0 ) = y'(0) iky(O), q(0) = y'(0) - iky(0). 1, by (12.5.16), if x 2 0, we deduce on taking absolute Since I eZkix 1 values in (12.5.19) that
. (12.5.23) 0
From (12.5.12) we may now deduce that q ( x ) tends to a limit as x -+ m, being of bounded variation. Thus q(m) exists if Im k >, 0. Concerning p ( x ) we assert that, if Im k
p(x) as x -+
00.
For since I e-zkit 1
=
=
> 0,
o(e-2kiz)
e2qt,
(12.5.24)
we have from (12.5.11) that
so that to establish (12.5.24) it will be sufficient to show that
388
12.
ASYMPTOTIC THEORY OF INTEGRAL EQUATIONS
This is evident on writing the left-hand side in the form
both terms on the right being of order o(e2qz). We deduce a general, though slightly incomplete, asymptotic expression for solutions of (12.5.1-2) when I m k > 0. Sincep(x) exp (2ki.r) 0 as x -P a, we have from (12.5.9-10) that --f
y ( x ) = -(2ki)-'q(o.)
and y'(x) = + q ( a ) e - t k x
r t k z
+
+o(rtkz)
o(e-tkx).
(12.5.25)
(12.5.26)
Provided that q ( a ) # 0, this solution is one of the type whose existence was asserted in (12.5.5-6). We proceed to verify that q ( a ) # 0 for at least one solution. If in (12.5.23) we take it that y(0) is zero we obtain
If therefore it so happens that
it will follow from (12.5.12) that
144 - d o ) I
15wb/(2n),
(12.8.27)
n
of which (12.8.10) is a weakened form. We conclude from this that there is at any rate one k, with bk, = n r p, , - 1 < pn < 1, such that y ( b , k,) = 0, provided that (12.8.10) holds. We proceed to find a better estimate of p, . Putting k = k , in (12.8.20) we obtain
+
I sin p n I
< exp (w/kn) - 1 < exp {bw/(nn- 1)) - 1 < exp {3wb/(2nw)) - 1,
(12.8.28)
as for (12.8.24). By (12.8.25-26), we deduce that
I sin pFLn I < (1019) {3wb/(2nm)) = 5wb/(3nm). By (12.8.21) we have, however, 1 sin p,
(12.8.29)
I 2 4 I p, 115, and so
4 I pn l/5 < 5wb/(3nm),
(12.8.30)
which proves (12.8.11), in sharper form. This completes the proof of Theorem 12.8.1 for the case that u(x) is real-valued, except that we have not proved the uniqueness of K, as described in the theorem. This we leave as a consequence of the modified argument to deal with the case of complex u(x), to which we proceed now. We put once more bk = n7r p, where I p I 1. We have then I exp ( - i k b ) I = I exp (-+) I e, and likewise I exp (ikb) I e. Hence from (12.8.18-19) we have
+
0 we can find an integer n and constants cl, ..., c, such that
If@) 3 -
0
c,u,(x)
1
dx
< E.
(12.10.1)
412
12.
ASYMPTOTIC THEORY OF INTEGRAL EQUATIONS
It may happen that (12.10.1) can be arranged by taking Er c,u,(x) to be a partial sum of a certain series, that is to say, we have an expansion m
(12.10.2) in the sense that (12.10.1) holds if n is large enough, the c, being independent of n and of e. I n this case, assuming such an expansion to exist for all f ( x ) E L2(0,b), the un(x) are said to form a “basis” in L2(0,b). I n particular, they form a basis if they are both complete and orthonormal, in that (12.10.3) for all p , q = 1, 2, ...; for in this case the best possible choice of the c,. in (12.10.1) is given by
(12.10.4) independently of n. Thus, for example, we showed in Section 8.6 that the eigenfunctions un(x) formed an orthonormal basis in a certain subspace ofL2(a,b), the set of functions described in Theorem 8.6.1. The relevant question is now the stability of the property of completeness, that is to say, given a set of functions { ~ ~ ( x in ) } L2(0, ~ b) 1 which are known to be complete in this space, and a second set {vn(x)}rin L2(0,b) which differ little from the u,(x), respectively, under what conditions can it be deduced that the {vn(x)}ralso form a complete set in L2(0,b). A simple criterion for this, proved in Appendix VI, states that if the set {un(x)}y is both complete and orthonormal, and if
(12.10.5) then the set { ~ ~ ( x is ) }also ~ complete; it is not here assumed that the 1 vn are also orthonormal. I n our first application of this result we take it as known that the orthonormal system un(x) = (2/6)Ii2 sin (nscrlb),
n = 1, 2,
...,
(12.10.6)
from a complete set in L2(0,b), that is, that the expansion given by a Fourier sine series holds in the mean-square sense. On the basis of this we have
12.10.
COMPLETENESS OF THE EIGENFUNCTIONS
413
<
0,
1;
{ m ( x , y)ISdx
=
1.
(12.10.13-14)
We have to show that there is a subdivision of (0, 1) by points (yt}, 0 = yo < y1 < ... < yl = 1, such that (12.10.15)
for Y = 0, ..., 1 - 1. From the assumed completeness of the system yn(x,y,) when Y = 0, that is, that of the system (12.10.6), there follows in turn its completeness for Y = 1, ..., 1, the latter giving the desired conclusion. T o show that (12.10.15) is possible we first choose an integer no such that
2
n-n0
r { y n ( s ,y ) - (2/l~)'/~ sin ( n ~ x / b )dx} ~< 1/8, 0
(12.10.16)
12.10.
COMPLETENESS OF THE EIGENFUNCTIONS
415
<
x implies u(x’) .(.). Here o(x) is of bounded variation over (a, 6 ) if and only if .(a) and o(b) are finite. T h e second case is that of a step function. I n the case of a finite interval (a, b ) we shall restrict this to mean a step function with a finite number of discontinuities. There will thus be a finite number of points in a x b between (in the strict sense) any two consecutive members of which U(X)is constant. In the case of a step function in an infinite interval, we permit an infinity of discontinuities, which are, however, to have no finite point of accumulation. T h e above definitions extend with little difficulty to the case in which u(x) is a square matrix, or rectangular matrix or vector, dependent on x. In saying that u(x) is of bounded variation over (a, b ) we mean simply that each entry in the matrix ~ ( x is ) separately a function of bounded variation in the previous sense. This notion will arise mainly in the special case in which o(x) is a square Hermitean nondecreasing matrix function; the term nondecreasing will mean that o(x’) - U(X) is positivedefinite, or at any rate non-negative definite, if x‘ > x. T h e entries on the principal diagonal will then be real-valued and nondecreasing functions in the ordinary scalar sense, and if they are of bounded variation, that is, bounded, then U(X)will be of bounded variation; this follows from the observation that for a non-negative definite matrix, no entry exceeds in absolute value the greatest diagonal entry.
>
<
{4&+1)
r=o
where the 7,.are arbitrary points with subject to m,ax I &+I - 5,
5,
I
< r], < 0,
. If as
TZ ---t
Q),
(1.2.3)
the sum (1.2.2) tends to a unique limit, for all sequences of subdivisions
1.2.
419
THE RIEMANN-STIELTJES INTEGRAL
satisfying (1.2.3) and for all choices of the 7, E [ f , , f,+,],then this limit is the Stieltjes integral of f ( x ) with respect to u(x), written J b f ( x )du(x), or Sb d u ( x ) f ( x ) , the order being important only in matrix ahalogs. In the case in which u(x) is a step function, the Stieltjes integral reduces to a sum. Theorem 1.2.1. In the finite interval [a, b] let f ( x ) be continuous and let u(x) be a step function, with a finite number of jumps. Then
+f(@ ( 4 4- o(b - 011.
(1.2.4)
Here the sum on the right is in fact a finite sum, being extended over points of discontinuity of u(x) in the interior of (a, b). The first and last terms are absent if u(x) is continuous at x = a or at x = b, respectively. - u(a)}, For the proof, the first term in (1.2.2), namely, f ( q 0 ) where a v0 El, clearly yields the first term on the right of (1.2.4) as n + m, since f(rl0) -+f(a) by continuity, and --+ a(a 0); here the limiting transition is as n ---t m, with qo and f1 functions of n. In a similar way, the last term in (1.2.4) arises from the last term in (1.2.2). If next x is a point of discontinuity of u(x) in a < x < b, suppose first that x is not a point of the subdivision (1.2.2), and denote by (5, 5') the interval of the subdivision containing x; suppose further that the subdivision is so fine that no other point of discontinuity of u(x) occurs in (5, f ' ) . I n this case we have in (1.2.2) a term of the form f(7) {u(S') - u(E)}, 7 E [ f , 5'1, which tends as n --+ 03 to the typical term in the sum in (1.2.4). If again x is a point of discontinuity, which is also a point of the subdivision, let 5, t',be the adjacent left and right points of the subdivision; we then get in (1.2.2) a pair of terms of the form f ( 7 ) {u(x) - ~ ( 5 ) ) f(7') (~(5') - a ( x ) ) , which yield in the limit as before f ( x ) {u(x 0) - u(x - 0)). We may complete the proof by showing that terms in (1.2.2) for which [t,,f,,,] does not contain a point of discontinuity of u(x) vanish. We pass to the existence of the Stieltjes integral in general. The main result is
{.(el)
<
,
’:}L({
formed in the same manner with a second subdivision We compare S with S’ by comparing both with a third sum
of ( a , b).
formed with the subdivision {t;};‘‘, comprising both the points {(,.} and the points {ti}.Write also 6 for the largest of the lengths of any of the intervals (5, , t,.+l)and (ti,(:+,) and w(6) =
max If(x’) - f ( x ) I,
I x’
subject to
-x
I
< 6.
(1.2.5)
We now write the sum S as a sum with respect to the joint subdivision {(:}. Noting that
45,+1>
-
45,)
=
2
eve;
{45;+J
-
45;,>
and substituting in (1.2.2) we may write S in the form
where r/? lies somewhere in that interval [(,, , (,+,I of the first subdivision which contains the interval [(;, (+ ;], of the joint subdivision. We have then
s - S“
2 If(%) -rc.l;,>
n”-1
=
c4+ ;!1,
p=0
-
a;,>.
Since r/,. and 7; both lie in the same interval [(,. , (,.+l] (1.2.5) that
it follows from (1.2.6)
< ~ ( 6 var ) (u(x); a, b}.
(1.2.7)
1.2.
THE RIEMANN-STIELTJES
INTEGRAL
42 1
Since the same argument applies to S’, we deduce that
IS
-
S’ I
du(x),
(1.2.1 1)
if the limit exists, and likewise (1.2.12)
Two cases in which the latter limit exists are relevant.
I.
422
SOME COMPACTNESS PRINCIPLES
Theorem 1.2.4. Let f ( x ) be continuous for all finite real x, and uniformly bounded, and let u ( x ) be of bounded variation over (-m, a ) . Then the integral (1.2.12) exists. Using an evident additive property of the integral, we have for a’ < a < b < b’,
so that to prove the existence of the limit (1.2.12) it will be sufficient to prove that
(1.2.13) as b, b‘ + 00, b < b‘. together with a corresponding result for the integral over (a’, a) as a + - m . By (1.2.10) and the assumed uniform boundedness of f ( x ) it will be sufficient for (1.2.13) to show that var ( u ( x ) ; b, b‘} + 0 as b + m . This is so since var { u ( x ) ; 0, b} tends monotonically to a finite limit as b + m, and since var (u(x); b, 6 ’ ) = var (u(x); 0, b’} - var (u(x); 0, b}
for
0
< b < b’.
The analogous result for the integral over (a’, a) is proved similarly. In connection with orthogonal polynomials we need
on
Theorem 1.2.5. --03
Let u(x) be defined, real-valued, and nondecreasing
< x < m, and for all positive integral n let, as x + m,
Then the integral
jmx m d u ( x )
(1.2.16)
-W
>
exists for all integral m 0. As in the case of the last theorem it is sufficient to prove that
J
b’
b
as b
3
m, a + - m ,
Jl,
x m d u ( x ) +O,
with b’
> b,
a’
xm du(x) -+ 0
< a.
(1.2.17-1 8)
1.3.
A CONVERGENCE THEOREM
423
We suppose that b > 1 and break up the interval (b, b') into intervals of the form (b, 2b), (2,4b), ..., with possibly a part of such an interval. Denote by M an upper bound for xm+l{u(m) - u(x)} for x 3 1. For b x 2b we have xm (2m/b)hm+l and so
<
0
for
Imh
< 0,
(11.2.7)
provided that f ( h ) is not merely a real constant. Since the zeros of p(h), q(h) have the separation property, the degrees of p(h) and q(h) will differ by at most unity. If q(h) is a constant, we shall b)/c for real a, b, c, for which the assertion is trivial. havef(h) = (ah Suppose then that q(h) has n zeros, denoted p1 < p2 < ... < pn , and
+
11.3.
SEPARATION PROPERTY IN MEROMORPHIC CASE
439
suppose first that p(A) is of degree n - 1, with zeros A,, ..., An-l , where < A, < /-L,.+~. T h e standard partial fraction formula gives here
p,.
(11.2.8)
Here we must observe that the coefficients p(p,.)/q’(p,.) all have the same sign, for p(p,.) and p ( ~ , . +will ~ ) have opposite signs by the separation property, while q’(p,.) and q’(p,.+,) have opposite signs since q(h) has only simple zeros. If, for example, the p(p,.)/q’(pr)are all positive, then (11.1.1-2) hold, with strict inequality, since I m (A - p,.)-l has the opposite sign to I m A ; if, of course, the coefficients in (11.2.8) are all negative, we get (11.1.1-2) with reversed signs. Suppose next that p(A) has the same degree as q(A), so that (11.2.8) must be supplemented on the right by a term p ( m ) / q ( m ) , meaning limA-,mp(h)/q(h).Since this term is a real constant, (11.1.1-2) are unaffected, and the same proof holds good. Finally, take the case in which p(A) is of degree one greater than q(A). This reduces to a previous case one if we consider l l f ( A ) = q(A)/p(A). For this case we can say that I m Ilf(A) has either always the same sign as I m A, or else always the opposite sign as Im A. On taking reciprocals these situations are interchanged, and we have that Imf(A) has either always the opposite sign to I m A, or else always the same sign. This completes the proof of Theorem 11.2.1.
11.3. Separation Property in the Meromorphic Case I n the case whenf(A) no longer is rational but is still meromorphic we can show that the “negative imaginary’’ property still implies a separation of its zeros and poles.
Theorem 11.3.1. Let f(A) be analytic, except for possible poles on the real axis, these poles having no finite limit-point, and let (11.1.1-2) hold (and so indeed (11.2.6-7) iff(A) is not a constant). Then the zeros of f(A), lying necessarily on the real axis, separate and are separated by the poles. T h e argument of (11.2.1-3) shows that f(A) can have no eomplex zeros; the same argument shows in fact that Imf(A) Gannot vanish for complex A, apart from the case when f(A) is a real constant. As before, f(A) cannot have a pair of zeros not separated by a pole,
440
11.
FUNCTIONS OF NEGATIVE IMAGINARY T Y P E
for if it did it would have an extremum for some real A, leading to a contradiction with (11.1.1-2). T h e hypothesis of two poles not separated by a zero likewise leads to a real extremum, and is therefore rejected. T h e above theorem may be applied to the proof of certain SturmLiouville separation theorems. We refer to other sources for the further development of the theory of “negative imaginary” functions, their general expression in such forms as (2.4.5), and the inversion of the latter, these being properties we have not appealed to.
APPENDIXI11
0rt hogonalit y of Vectors
111.1. The Finite-Dimensional Case We use frequently the simple observation that if the rows of a square matrix are mutually orthogonal, and not zero, then the columns are likewise orthogonal, with suitable weights. This is, with a slight transformation, a well-known property of an orthogonal matrix. However, we give a direct proof.
Theorem 111.1.1. Let y r S , r, s = 0, ..., m - 1 be orthogonal according to m-1
-
~ a r Y r s Y r= t b% s, t 9
...,m - 1 ,
= 0,
(111.1 . l )
r=o
where the ur , ps are real and positive. Then (I1I. 1.2)
I t follows from the orthogonality (111.1.1) that the m vectors
yoa, ...,y n , - l , s , ( s = 0, ..., m - I), are linearly independent. Thus an
arbitrary vector uo , ...,
may be expressed in the form
(111.1.3)
here the p p are normalization factors, and the Fourier coefficients vup are to be found. Multiplying by urLynrand summing over n we get
441
442
111.
ORTHOGONALITY OF VECTORS
by (111.1.1). Hence, substituting for wp in (111.1.3),
p=o
q=o
Here the un are arbitrary, and (111.1.2) may therefore be derived by comparing coefficients of the uq . The Parseval equality (111.1.4)
may be verified on substituting for the u, from (111.1.3), and using (111.1.1).
111.2. The Infinite-Dimensional Case In the case m = it is not possible to deduce (111.1.2) from (111.1. I), but some deductions can nevertheless be made.
Theorem 111.2.1. Let ur , pr , Y = 0, 1, 2, let the y r 8 ,Y , s = 0, 1, 2, ..., satisfy *
-
ZySrytTp;' = aSta;l,
Then
..., be real and positive,
s, t = 0,1,
... .
and
(111.2.1)
r-0
For an arbitrary sequence u,, , u1 , ... , satisfying
define (111.2.4)
111.2. THE INFINITE-DIMENSIONAL CASE
443
Then there holds the Parseval equality m
m
We prove first that the Parseval equality is true for finite sequences of the form u o , u l , ..., u, ,O,O,... . Defining, in accordance with (111.2.4), =
vrn
we have
2
auUdu7
(111.2.6)
9
P=o
2I
v r n 1 2 P;'
=
r-0
2 22 P;'
_-
apaau,ugyuvyar
u-Oa=O
r-0
P=O a-0
r-0
the last series being absolutely convergent ; this last statement follows from (111.2.1) for s = t together with the Cauchy inequality. Using (111.2.1) we deduce that m
n
r=O
u-0
(111.2.7)
in confirmation of (111.2.5) for this case. Let us now prove (111.2.2). We take up = y p 8 , for 0 Q p Q n, up = 0 for p > n. From (111.2.6) we have then n
P-0
Taking on the left of (111.2.7) only the term for
I =
s we have
n
Ivm12Pg1 < Z a u I y u s l a= v m . u=o
Hence v,
< p a , that is to say,
2% < n
u=o
IYP. 12
P S .
Since n is arbitrary, we have (111.2.2) on making n -+
00.
444
111.
ORTHOGONALITY OF VECTORS
Letting now uo , u1 , ... ; be any sequence satisfying (111.2.3), we pass to the proof of (111.2.5) for the general case. With the notations (111.2.4), (111.2.6) we have vrn+ w, as n -00, (111.2.8) in view of (111.2.2-3) and the Cauchy inequality. Furthermore, for O 0, N
m
r=O
and so, making
7tl-
Making now N
--t
p=n+l
Q]
Q]
and using (111.2.8),
we have
from which we draw the conclusion, a sharpening of (111.2.8), that W
~1wr-vrn~2p;'+0
as
n+m.
(III.2.9)
r=O
We may now prove (111.2.5) by making n + m in (111.2.7); it is necessary to show that
(111.2.10)
111.2. THE INFINITE-DIMENSIONAL CASE
445
and so (111.2.10) follows in view of (111.2.9), and in view of the fact that
2 I vrn I I m
r=o
vr
- vrn I P;'
-
0;
the latter follows from (111.2.9), (111.2.7) together with the Cauchy inequality, This completes the proof. Together with the Parseval equality (111.2.5) we have an associated expansion theorem. Theorem 111.2.2.
Under the assumptions of Theorem 111.2.1, c=
u, = z v D y n p p ; ' ,
n
= 0,
I , 2, ... .
(111.2.11)
p=O
With the notation (111.2.6) we have, if m >, n,
to be verified by substituting for vpUpm by (111.2.6) and using (111.2.1). On making m -+ 00, we obtain (111.2.1 l), subject to it being proved that
This follows from (111.2.9), (111.2.1) and the Cauchy inequality. In connection with polynomials orthogonal on the real axis we note also a partial extension of Theorem 111.2.1 from discrete sums to Stieltjes integrals involving jumps. Theorem 111.2.3. Let ~ ( h ) --oo , < h < m, be non-decreasing and have a jump l/p' > 0 at X = A'. Let the continuous functions y,,(h), n = 0, 1, 2, ... , be such that
for certain positive a, , the integrals being absolutely convergent. Then m
446
111.
ORTHOGONALITY OF VECTORS
We modify (111.2.6) to
getting
From this we deduce that
The spectral choice up = y,(A’) together with the limiting transition n + m then gives the result as before.
APPENDIXIV
Some Stability Results for Linear Systems
IV.1. A Discrete Case We prove here for solutions of difference or differential equations some conditions which ensure convergence at infinity, or which yield bounds for large values of the independent variable. We start with the discrete analog of a well-known theorem on the convergence of solutions of differential equations.
Theorem IV.l.1. An,
Let the sequence of k-by-K matrices n = 1,2, ...;
An = (anre),
r, s = 1, ...,K,
satisfy (IV.1.1)
where (1V.1.2) r-1 s=l
Then the solutions of the recurrence relations x,+~ - x,, = A,x,,
,
n
=
1,2, ...,
(IV.1.3)
where x, is a k-vector, converge as n 3 m. If in addition the matrices (E A,) are all nonsingular, then limn+mx, # 0, unless all the x, are zero. Writing x, ,r = 1, ..., k, for the entries in x, and defining its norm by
+
2I k
xn
=
r-1
447
xnr
I
(IV.l.4)
Iv.
448
SOME STABILITY RESULTS FOR LINEAR SYSTEMS
we have from (IV.l.3) that
and so, summing over
Y,
0. Choosing z' sufficiently small, the total of all the remaining jumps of aj(x) will be less than Q E , and a,(x) will have the above property. We have now that
Using the bounds (IV.4.7), (IV.4.8) we deduce that
taking it that
E
< 1 so that the last logarithms are real. Using the results
IV.5.
THE EXTENDED GRONWALL LEMMA
455
3 0, log (1 + 7) < 7, and that log (1
1
B C{U(X)
- a(a)>,
and so tends to zero as E -P 0. We then derive (IV.4.6) from (IV.4.10), which completes the proof.
IV.5. The Extended Gronwall Lemma T h e following result gives the direct extension to Stieltjes integrals of the fundamental lemma of Gronwall (cf. Bellman, “Stability Theory,” p. 35).
<
0, are constants. P(4
p
< co exp {c1[+)
Then - .(a)l).
(IV.5.2)
T h e result will obviously be established if we prove that for arbitrary > 1 and a x b we have
<
.
(IV.5.3)
For any chosen p > 1, this will be true for x = a by (IV.5.1), and in a right-neighborhood of a, by continuity. If (IV.5.3) does hot hold, for the p in question, for all x E [a, b], let a’ be the greatest number in (a, b] such that (IV.5.3) holds for a x < a’. Substituting x = a’ in (IV.5.1), and replacing the integral by a sequence of approximating sums we have
1; the case r = 1 has just been disposed of, and may be ignored. T h e set of admissible x is in fact given by (V.1.5-6) with c1 = ... = c,-~ = 0, and the conclusion follows from (V.1.7). T h e expression (V.1.8) for A,, has the disadvantage of involving a knowledge of xl, ..., xrP1 . This is remedied in the following alternative, though more complicated, expression for A,.
0, or B 2 0, in the sense that the associated quadratic form x*Bx is definite or semidefinite and non-negative. Let us write, if B 2 0, 11 B = min{x*Bx I x*x = l}, (V.2.3) 2
+
so that this is the least of the eigenvalues of B, assumed to be all non0 to include the case B > 0 we have negative. Using the phrase B
Theorem V.2.2.
If A, B are Hermitean and B 2 0, then &(A
For x such that x*x
=
+B)
-
&(A)2 I I B IIt.
1 we have now
x*(A + B ) x --*Ax
and so m(A
(V.2.4)
+ B , y , , ...,
Y7-1)
2 1 1 B Ilt,
= x*Bx
- m(A,y,1 ...,Y7--1)
II B IT.
T h e same result now follows for A,. by means of (V.1.10). If now A(t) is an increasing, or nondecreasing, Hermitean matrix function in the sense that A(t’) - A(t) is positive or non-negative definite when t’ > t , we may conclude that its eigenvalues A,.(t) are increasing, or nondecreasing, functions of t in the ordinary sense. As a continuous variant of the last result we have Theorem V.2.3. I n a real t-interval let A(t) be a differentiable Hermitean matrix function, whose derivative A’(t) is positive-definite. Then the A,.(t), the eigenvalues of A(t),are increasing functions of t . For any fixed t’ and variable t“ + t’ we have A(t”) - A(t’)= (t”
+
-
+
t‘) A’(t’)
O(t”
- t’)
as t” + t’ 0. From this, and the fact that A’(t’) > 0, it is easily deduced that A(t”) > A(t’) for sufficiently small t” - t’, so that A(t) is an increasing function of t . T h e result of the theorem now follows from the last theorem.
v.3.
46 1
A FURTHER MONOTONICITY CRITERION
V.3. A Further Monotonicity Criterion I n connection with matrix Sturm-Liouville theory we need an extension of Theorem V.2.2, under which B need not be fully positive-definite, but only on the linear subspace formed by the eigenvectors of A for the eigenvalue in question. We first consider the comparison of two matrices, and then give a continuous version. Theorem V.3.1. Let A, B be Hermitean and A' an eigenvalue of A , possibly multiple, so that &(A)= A', For some p
r = u, u
> 0, and all x with Ax x*Bx
=
+ 1, ...,w.
(V.3.1)
X'x let
3 px*x.
(V.3.2)
If A' is the greatest of the eigenvalues of A, that is, if u = 1 in (V.3.1), then &(A + B ) > A,(A), r = u, u + 1, ..., v. (V.3.3) If X is not the greaiest of the eigenvalues of A, let X'be the next greater, and let p' denote 11 B 11 as defined in (V.2.1). If p'2
< p(A"
- A' - p),
Pz
2
then (V.3.3) holds. Let PI =
2
xrxT*,
=
1,(A) < A '
xrxT*,
d7(A)=d'
(V. 3.4)
P,
=
2
x,x:.
& ( A )> A '
We assert that, for r = u, ..., v, h,(A
+ B ) >, m$
{x*(A
+ B) x
x * x = 1,
+
x = (Pz P,) x ) . (V.3.5)
This follows from Theorem V.1.4, if in place of Y we have z, and in ..., x k . T h e requirements x,*x =0, place of y r + l ,..., yk take s = z, + 1 , ..., k imply that Plx = 0; these requirements are to be omitted if z, = k, that is to say, if A' is the lowest eigenvalue. Furthermore, we have P, Pz P , = E, the unit matrix, by (V.l.2), the eigenvectors forming a complete orthonormal set, so that if P , x = 0 then x = (Pz P,) x. This justifies (V.3.5). We first dispose of the trivial case in which A' is the greatest eigenvalue, so that P, = 0, and in (V.3.5) we have x = P z x , and so Ax = X'x.
+ +
+
v. EIGENVALUES
462
OF VARYING MATRICES
+
+
It then follows from (V.3.2) that x*(A B ) x 3 x * A x px*x = (A' p ) x*x, and so from (V.3.5) that &(A B ) A' p, in verification of (V.3.3) for this case. For the more general case that P, # 0 we have, if x = (P2 P,) x,
+
+
+
+
x*(A
+ B ) x = x*(PZ+ P3)A(P, + P,) x + x*PzBP2x + x*P,BP, x + x*P,BPz x + x*P3BP3x ,
(V.3.6) where we have used the facts that P: = Pz , P$ = P3 . Here we note that APzx = A X?X?X = A'PZx,
2
Ar(.4)-aa
so that
x*PZBPzx> ~ x * P , x ,
x*P2APZx= A'x*P,x,
using the fact that Pi
=
P 2 . In a similar way we have
x*P3AP3x > A"x*P,x,
and, using (V.2.1) with pf
=
X'P,AP,X
= x*P,AP,x
= 0,
11 B 11,
I x*P,BP,x 1
< p'x*P,x.
Using these results in (V.3.6) we obtain x*(A
+ B ) x > (A' + p ) x*P,x + (A"
+ x*PZBP3x+ x*P3BPZx.
- p') x*P,x
(V. 3.7)
We introduce a notation for the length of an arbitrary column matrix, writing 1 z 1 = 1/(z*z) 3 0. For any other column matrix y we have the Cauchy inequality I z*y 1 I z I . I y I. We have then
0 and 0 < t, - to < E, we shall have 11 A(t,) - A(t,) /I ~ ' ( t , to), with the interpretation (V.2.1). This yields the situation of Theorem V.3.1, with
A,.(t,) = A,, for 0 < t , - to < E and the r-values for which h,(t,) = A,. If A, is not the greatest eigenvalue, we denote by Ah the next greater eigenvalue of A(t,), and the same conclusion follows, provided that, according to (V.3.4),
A,(t,)
V'2(tl
- t,)2
< &V(t,
- to) [A;
- h, - 1 24tl
- t0)L
This is plainly satisfied for some E' > 0 and 0 < t, - to < E'. We deduce once more that A,(t,) > A,(t,) = A, for t , in some rightneighborhood of to . T h e corresponding statement that &(t,) < A,(t,)=h, for t , in some left-neighborhood of to may be proved by applying the same argument to -A(-t).
V.4. Varying Unitary Matrices We now pass to the situation in which we are given a matrix O(t), of the kth order, which for to t t , is unitary, in that e(t) e*(t) = O*(t) d ( t ) = E, and which is also continuous in t , in that all its entries are continuous. We denote its eigenvalues by ~ , . ( t ) ,r = 1, ..., k, not necessarily all distinct and written a number of times according to multiplicity. T h e w,.(t) lie necessarily on the unit circle, and we need results giving conditions under which they move monotonically on the unit circle as t increases. Two differences emerge when we compare this situation with that of the eigenvalues of a varying Hermitean matrix A(t).Whereas a Hermitean matrix A(t) defined in connection with a differential equation may become infinite, and therewith also some of its eigenvalues, even though the differential equation exhibits no irregularity, a unitary matrix and its eigenvalues are necessarily finite; this was, in Chapter 10, a motive for introducing such matrix functions. On the other hand, a difficulty arises in connection with the identification of the eigenvalues. For a
<
0, chosen so that exp (ia) is not an eigenvalue for t‘ t < t’ 6 , the ~ ~ (are t )to be numbered in the same order when
+
0, in the matrix sense of this inequality.
List of Books and Monographs
(References to these items in the Notes will be abbreviated, usually to the author’s name only. Other items will be cited with full bibliographical details.) AHIEZER,N. I. (ACHIEZER),“Lectures on the Theory of Approximation.” MoscowLeningrad, 1947; German ed., Akad.-Verlag., Berlin, 1953. AHIEZER, N. I., and GLAZMAN, I. M. (GLASMA”), “Theorie der linearen Operatoren in Hilbert Raum.” Moscow, 1950; German ed., Akad.-Verlag., Berlin, 1960. BECKENBACH, E. F., and BELLMAN, R. E., “Inequalities.” Springer, Berlin, 1961. BELLMAN R. E., “Stability Theory of Differential Equations.” McGraw-Hill, New York, 1953. BELLMAN, R. E., “Introduction to Matrix Analysis.” McGraw-Hill, New York, 1960.
L., “Theorie der Differentialgleichungen.” Berlin, 1926. BIEBERBACH, BIRKHOFF, GARRETT,and ROTA,G.-C., “Ordinary Differential Equations.” Ginn, Boston, Massachusetts, 1962. BIRKHOFF, GEORGE D., “Collected Works,” Vol. I. New York, 1950. CODDINGTON, E. A,, and LEVINSON, N., “Theory of Ordinary Differential Equations.” McGraw-Hill, New York, 1955. COLLATZ,L., “The Numerical Treatment of Differential Equations.” Springer, Berlin, 1960. COURANT, R., and HILBERT, D., “Methods of Mathematical Physics,” Vol. 1. 2nd German ed., Berlin, 1931; English ed., Wiley, New York, 1953. DOLPH,C. L., Recent developments in some non-self-adjoint problems of mathematical physics, Bull. Amer. Math. SOC.67 (1961), 1-69. FORT,T., “Finite Differences and Difference Equations in the Real Domain.” Oxford. Univ. Press, London and New York, 1948. GANTMAHER, F. R. (GANTMACHER), and KRE~N, M. G., “Oscillation Matrices, Oscillation Kernels, and Small Vibrations of Mechanical Systems.” 2nd Russian ed., MoscowLeningrad, 1950; German ed., Akad.-Verlag., Berlin, 1960; English ed., USAEC translation 4481, 1961. 478
BOOKS AND MONOGRAPHS
479
GEROAIMUS, YA. L., “Theory of Orthogonal Polynomials.” Moscow, 1958; English ed., “Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval.” Consultants’ Bureau, New York, 1961; or “Polynomials Orthogonal on a Circle and Interval.” Pergamon, New York, 1960.
~ , “Toeplitz Forms and Their Applications.” Univ. of GFENANDER, U., and S Z E C G., California Press, Berkeley, California, 1958. HANNAN, E. J., “Time Series Analysis.” Wiley, New York, 1960. INCE,E. L., “Ordinary Differential Equations,” 4th ed. Dover, New York, 1953. KAMKE,E., “Differentialgleichungen reeller Funktionen.” Teubner, Leipzig, 1930. KARLIN,S., and S Z E CG., ~ , On certain determinants whose elements are orthogonal polynomials, J. Anal. Math. 8 (1960/61), 1-157. KRE~N, M. G., and KRASNOSEL’SKI~, M. A., Fundamental theorems on the extension of hermitian operators and certain of their applications to the theory of orthogonal polynomials and the problem of moments, Uspekhi Mat. Nauk 2 (1947), 60-106. KRE~N, M. G., The ideas of P. L. Cebysev and A. A. Markov in the theory of the limiting values of integrals and their further development, Uspekhi Mat. Nuuk 6 (1951), 3-120; Amer. Math. SOC.Transt. (2) 12 (1959), 3-120. KRE~N, M. G., and REHTMAN, P. G., Development in a new direction of the CebySevMarkov theory of the limiting values of integrals, Uspekhi Mat. Nuuk 10 (1955), 67-78; Amer. Math. SOC.Transl. ( 2 ) 12 (1959), 123-136.
B. M., Appendices I-V to Russian ed. of Part I of “Eigen-Function Expansions” LEVITAN, by E. C. Titchmarsh. Moscow-Leningrad, 1960. MORSE, M., “Calculus of Variations in the Large.” h e r . Math. SOC.Colloquium Publications, Vol. 18, New York, 1934. NA~MARK, M. A. (NEUMARK), “Linear Differential Operators.” Russian ed., Moscow, 1954; German ed., Akad.-Verlag., Berlin, 1960. N A ~ M A RM. K ,A., Investigation of the spectrum and the expansion in eigenfunctions of a non-self-adjoint differential operator of the second order on a semi-axis, Trudy Moskow. Mat. ObE. 3 (1954), 181-270; Amer. Math. SOC.Trunsl. ( 2 ) 16 (1960), 103-193. POTAPOV, V. P., The multiplicative structure of ]-contractive matrix-functions, Trudy Moskow. Mat. ObP. 4 (1955), 125-236; Amer. Math. SOC.Transl. ( 2 ) 15 (1960), 131-243. RIESZ,F., and SZ~KEFALVI-NAGY, B., “Leqons d’analyse fonctionelle.” Budapest, 1953. SHOHAT, J., and TAMARKIN, J. D., “The Problem of Moments,” Math. Surveys No. 1. Amer. Math. SOC.,New York, 1943, 1950. STONE,M. H., “Linear Transformations in Hilbert Space and Their Applications to Analysis.’’ Amer. Math. SOC.Colloquium Publications, Vol. 15, New York, 1932.
480
BOOKS A N D MONOGRAPHS
SZECO,G . , “Orthogonal Polynomials.” Arner. Math. SOC. Colloquium Publications, Vol. 23,New York, 1939;2nd. ed., 1959.
TITCHMARSH, E. C., “Eigenfunction Expansions Associated with Second-Order Differential Equations,” Part I. Oxford Univ. Press, New York, 1946; 2nd. ed., 1962; Part 11, Oxford Univ. Press, New York, 1958.
TITCHMARSH, E. C., “The Theory of Functions.” Oxford Univ. Press, New York, 1932; 2nd ed., 1939. TITCHMARSH, E. C., “Theory of Fourier Integrals,” Oxford Univ. Press, New York, 1937. WALL,H. S., “Analytic Theory of Continued Fractions.” Van Nostrand, Princeton, New Jersey, 1948. WIDDER, D. V., “The Laplace Transform.” Princeton Univ. Press, Princeton, New Jersey, 1946.
Notes
Section 0.1 Some discussion of the discrete boundary problem (0.1.5), (0.1.8), and of variational and other aspects, is given on pp. 142-146 of Bellman’s “Matrix Analysis.” Practical numerical aspects of the replacement of the boundary problem for a differential equation by that for a difference equation are treated in works such as that of L. Collatz, in Chapter I11 of the cited book. For the use of the discrete approximation to establish the eigenfunction expansion for the differential equation case see Levitan, Appendix I to the Russian edition of Titchmarsh’s book, or M. PLANCHEREL, Le passage A la limite des Cquations aux diffkrences aux Cquations diffbrentielles dans les problbmes aux limites, Bull. Sci. Math. 46 (1922), 153-160, 170-177;
the matter is also referred to in Fort’s book. Although we shall not reproduce this argument in this book, we use the process to establish the expansion theorem for a certain mixed discrete-continuous recurrence relation, generalizing that associated with complex Fourier series; see Section 2.10.
Section 0.2 Concerning this type of wave propagation, see for example J. A. STRATTON, “Electromagnetic Theory.” McGraw-Hill, New York, 1941, Chapter 5 and Problems.
Section 0.4 See for example J. G. TRUXAL, “Control Engineer’s Handbook.” McGraw-Hill, New York, 1958.
48 1
48 2
NOTES
Section 0.7 For the further theory of the probabilistic model see the notes to Section 5.7 and the references given there. See also I. J. GOOD,Random motion and analytic continued fractions, R o c . Cambridge, Phil. SOC. 54 (1958), 43-47.
Section 0.8 Sturm-Liouville theory with a parameter in the boundary conditions has been treated by a number of writers; in particular see G . W. MORGAN, Some remarks on a class of eigenvalue problems with special boundary conditions, Quart. uppl. Math. 11 (1953). 157-165, W. F. BAUER, Modified Sturm-Liouville systems, ibid. 272-283,
R. L. PEEK,Jr., A problem in diffusion, Ann. of Math. (2) 30 (1929), 265-269, E. HILLE,Note on the preceding paper by Mr. Peek, ibid. 270-271.
There are many more general investigations, relating to more general boundary conditions or side conditions, systems of higher order, and so on. For such work and further references see J. D. TAMARKIN, Some general problems of the theory of ordinary linear differential equations and the expansion of an arbitrary function in series of fundamental functions, Math. 2. 27 (1927), 1-54,
R. E. LANCER, A theory for ordinary differential boundary problems of the second order 53 (1943), 292-361, and of the highly irregular type, Trans. Amer. Math. SOC.
L. A. D I K I ~On , boundary conditions depending on an eigenvalue, Uspekhi Mat. Nauk 15 /1960), 195-198,
J. ADEM, “Matrix differential systems with a parameter in the boundary conditions, Quart appl. Math. 17 (1959), 165-171, H. J. ZIMMERFIERG, Two-point boundary conditions linear in a parameter, Pacific J. Math. I2 (1962), 385-393.
In the Sturm-Liouville case, it is clear that the presence of the second derivative in the boundary conditions may be eliminated by means of the differential equation, at the cost of introducing the spectral parameter. For another approach to such problems see R. V. CHURCHILL, Expansions in series of non-orthogonal functions, Bull. Amer. Math. SOC.48 (1942), 143-149.
We discuss in Chapter 8 and the notes for Section 8.1 various extended forms of Sturm-Liouville theory in which the presence of the parameter
483
NOTES
in the boundary conditions in linear fashion forms part of a wider generalization which allows for discontinuities within the basic interval. For the case of Sturm-Liouville theory with a finite number of interface conditions a thorough investigation is due to W. C. Sangren, cited under the notes for Section 11.8. T h e condition that the matrix in (0.8.6), if constant, be symplectic may be ensured by multiplication by a constant factor, if it has positive determinant; this does not of course apply for higher dimensions. In this case orthogonality relations can still be set up if the symplectic property fails.
Section 1.5 T h e term “spectral function” occurs in the literature in three senses. I n that used in this book it gives a weight-distribution on the real axis with the property of inverting the definition of the Fourier coefficient to yield the function being expanded in eigenfunctions, as (1.5.5) is inverted by (1.5.6). One may also demand that (1.5.6) be inverted by (1.5.5), for a suitable class of a@). T h e definitions may be most simply illustrated in the case of the continuous analog of the recurrence relation of this chapter, that is in the case of the differential equation y’ = ihy, subject to y(0) = y(1). T h e expansion theorem, that of complex Fourier series, asserts that if for any well-behaved f ( x ) we define
v(h) = f f0( t ) e x p ( i h t ) d l , then
-00
where the spectral function ~ ( his) in this case the greatest integer not exceeding h , / ( 2 ~ T) .h e “dual orthogonality,” corresponding to (1.5.3), is now the formal result that
1
W
-m
exp ( i h ~exp ) ( i ~ td+) )
= a(x - t ) ,
the right-hand side being the Dirac delta function. T h e spectral function just defined has the orthogonal property that the relationship between f ( x ) and a(h) is reciprocal, isometric, and onto as between f ( x ) E L2(0,l), on the one hand, and the set of v(h) such that J” I v(h) l2 &(A) is finite, on the other, the latter being effectively -W
484
NOTES
the set of sequences of summable square, in view of the Riesz-Fischer theorem (Riesz and Sz.-Nagy, Chapter 2, or Titchmarsh, “Theory of Functions,” Chapter 13). For a distinct use of the term “spectral function” Iet us define rather the “spectral kernel” T ( X , y; A) by J O
where ~ ( his) as previously. Then, under suitable restrictions,
This spectral kernel may be defined as a step function with jumps at the eigenvalues, the jump being the product of two eigenfunctions, associated with the particular eigenvalue, the eigenfunctions being normalized in the mean-square sense, and in our present case using the complex conjugate of one of them. T h e same construction is important in Sturm-Liouville cases (see for example Levitan, Appendices to Titchmarsh’s book). For a similar construction in connection with partial differential equations, see for example F. J. BUREAU,Asymptotic representationof the spectral function..., 3. Math. Anal. Appl. 1 (1960), 423-483.
T h e third use of the term “spectral function” concerns the integral operator defined by the “spectral kernel” just introduced. We define a family of operators EA by
so that, formally at least, - i f ’ ( x ) = Jrn h d E a f ( x ) . For further informa--m tion on such “resolutions of the identity’’ we refer to the books of Ahiezer and Glazman or of Stone, or to papers of Naimark, such as his Extremal spectral functions of a symmetric operator, Izw. Akod. Nauk SSSR, Ser. Mat.
11 (1947), 327-344,
or R. C. GILBERT, The denseness of the extreme points of the generalized resolvents of a symmetric operator, Duke. Math. J. 26 (1959), 683-691.
NOTES
48 5
I t should be mentioned that in any of these senses the spectral functions (though not in general the orthogonal spectral functions) form a convex set, containing with any two such functions also their arithmetic mean with any non-negative weights. We may therefore distinguish extremal spectral functions lying, so to speak, on the boundary of this set, not representable as the arithmetic mean of other spectral functions. This notion appears in particular in the moment problem, and the convexity is reflected geometrically here and in the case of differential equations by the association of spectral functions with points of circles in the complex plane.
Section 1.6 We emphasize that the (somewhat overworked) term “characteristic function” will not be used in this book in the sense of an eigenfunction, but in the sense of a certain meromorphic function of the spectral parameter, having poles at the eigenvalues. A rather similar use of the term is made by Naimark in his book, p. 240, and by K. Kodaira in two important papers on differential equations [Amer. J. Math. 71 (1949), 921-945; 72 (1950), 502-5441. A distinct though not unrelated usage is followed in the definition of characteristic functions or matrix functions in a functional-analytic context by LivBic. See M. S. BRODSKI~ and M. S. LIVSIC,Spectral analysis of non-self-adjoint operators and intermediate systems, Uspekhi Mat. Nauk 13 (1958), 3-85, A . V. STRAUS, Characteristic functions of linear operators, Doklady Akad. Nauk SSSR 126 (1959), 514-516,
or, for a brief account, the book of Ahiezer and Glazman.
Sections 1.7-8 These form analogs of problems of inverse Sturm-Liouville theory, in which a coefficient-function or “potential” in a second-order differential equation is to be recovered, given either the spectral function, or alternatively given two sets of eigenvalues corresponding to two given boundary conditions at one end, the boundary condition at the other end being fixed. See Sections 4.6-7, 5.2, 7.4, 12.4, and the Notes to Section 12.7.
Section 1.10 T h e term “moment-problem” most commonly refers to problems
486
NOTES
concerning moments of powers on the real axis, that is to say, the determination of ~ ( hfrom ) the equations jh"dT(h)
= pn ,
n
= 0,1,
...,
the pn being given; the integral may extend over the whole axis (-m, m) (Hamburger problem), or over (0, a)(Stieltjes), or over (0, 1) (Hausdod); here it is mainly the Hamburger problem which is of interest in this book, though this does not exclude .(A) being constant on the negative real axis. One way of viewing the moment problem is that the moments define a scalar product of any two polynomials f ( h ) , g(X); the expression
j m f(4go d T ( 4 -W
involves only the moments, and so may be evaluated without knowledge of ~ ( h )Completing . this set of polynomials to a Hilbert space, we study the symmetric operator defined on polynomials by the mapping f ( h ) + hf(h). For this approach, due to Liviic, Krein, and Krasnosel'skii, see the cited monograph of M. G. Krein and M. A. Krasnosel'skii. In a similar way, for the problem of this section, we may suppose known the values of J"--m ( A - a,)-' &(A), where the a , are given but ~ ( h is) unknown, but is to be nondecreasing; the a, are to lie in the upper half-plane and in the simplest case are all distinct. These moments again determine a scalar product and so a pre-Hilbert space of rational functions with poles at the a,. T h e operator given by multiplication by h will be symmetric, with domain including those rational functions with at most simple poles at the a, which vanish to order O(h-2) as
x+
00.
Similar ideas apply to the trigonometric moment problem and its continuous analogs. For multivariate extensions see A. DEVINATZ, On the extensions of positive definite functions, Acta Moth. 102 (1959), 109-134.
where a connection is found with the work of N. ARONSZAJN, The theory of reproducing kernels, Trans. A m y . Math. SOC.68 (1950), 337-404.
Additional references on the ordinary power moment problem are given in the notes to Section 5.10, and on the trigonometric problem in the notes to Section 7.5.
487
NOTES
Hilbert spaces of analytic functions also occur in the work of L. DE BRANGES, Some Hilbert spaces of entire functions, IV, Trans.Amer. Math. SOC. 105 (1 963, 43-83,
where other references are given. T h e Pick-Nevanlinna problem consists in finding a function f(h), analytic in I m h > 0, Imf(h) having a fixed sign there, to take assigned values at an infinite sequence of points in I m h > 0. Imposing the first m of these conditions, and making m increase, there results a recurrence relation, leading to a limit-point, limit-circle classification; this classification is the analog of that obtaining in Sturm-Liouville theory when the basic interval is extended to infinity, or in the threeterm recurrence situation of Chapter 5. I n the present case, these recurrence relations suggest analogs for differential equations which involve the spectral parameter in fractional-linear form. See H. WEYL,Uber das Pick-Nevanlinna’sche Interpolations-problemund sein infinitesimales Analogon, Ann. of Math. (2) 36 (1939, 230-254.
See also Krein’s monograph, “The ideas of P. L. Cebysev that of Beckenbach and Bellman.
...,
”
and
Section 2.2 For recent results and bibliography on Blaschke products see G. T. CARGO, Angular and tangential limits of Blaschke products and their successive derivatives, Canad. J. Math. 14 (1962), 334-348, A. A. GOL’DBERG, Notes on Blaschke’ derivatives for a half-plane, Ukrain. Mat. Zh. 1 1 (1959), 210-213.
Section 2.3 Reasoning from the uniform boundedness of a family of spectral functions to the existence of a limiting spectral function is a device to be employed later in connection with orthogonal polynomials(Section 5.2), and is standard usage in the topic of Sturm-Liouville theory on a half-axis (Section 8.12).
Section 2.5 I n Theorem 2.5.1, if the real axis be transformed to the unit circle, we have to deal with the derivative of a Blaschke product, and the radial limit of this derivative; see the reference just made to the paper of G. T. Cargo.
488
NOTES
Section 2.7 In view of the orthogonality (2.7.l), the rational functions qn(h) result from applying the process of orthogonalization to the functions (1 - ihFn)-l, the orthogonality interval being the whole real axis with a constant weight-function. See 0. SzAsz, “Collected Works.” Cincinnati, 1955.
T h e same orthogonality may be applied to what we might view as a dual expansion theorem, in which a(h), defined on the real axis, is to be expanded in a series of the ~ ~ ( hLikewise, ). we may consider v(h) as a given meromorphic function to be expanded in such a series. For similar investigations see E. LAMMEL, Uber Approximation meromorpher Funktionen durch rationale Funktionen, Math. Ann. 118 (1941), 134-144.
Section 2.10
If we restrict the expansion to that of a function defined over the continuous range, here denoted by - c x 0, we make contact with, though without including, an investigation of
<
0. This forms a very special case of a theory of products of matrix factors with these properties, allowing also fractional-linear factors (A,h B,) (C,h Dn)-l, and allowing infinite products, discrete, continuous, or mixed. T h e basic work in the field is the monograph of V. P. Potapov, listed in the general references, which underlies all our discussion.
+
+
+
Section 3.2 Concerning the symplectic group see the book of H. Schwerdtfeger, “Introduction to Linear Algebra and theTheory of Matrices” (Groningen, 1950), or that of C. Chevalley, “Theory of Lie Groups, I” (Princeton
NOTES
489
Univ. Press, Princeton, New Jersey, 1946). Analytic aspects are taken UP by C. L. SIEGEL, Symplectic geometry, Amer. J. Math. 65 (1943), 1-86.
See also the notes for Section 10.1.
Section 3.3 Isotropic subspaces with respect to an indefinite metric are considered by A. I. Mal’cev, “Foundations of Linear Algebra” (Moscow-Leningrad, 1948), Chapter 9. See also the references to the work of V. A. YakuboviE on the symplectic group given in the Notes to Section 10.1.
Section 3.5 Transformations of the plane leaving area invariant are discussed by H. S. M. COXETER, “Introduction to Geometry.” Wiley, New York, 1961,
such linear transformations being termed “equi-affine”; the special case of a shift parallel to a fixed line, of amount proportional to the distance from it, is a “shear.” T h e term “symplectic transvection” is used by E. Artin, in “Geometric Algebra” (Interscience, New York, 1957).
Chapter 4 T h e theory of orthogonal polynomials is usually developed starting from the orthogonality; the latter is usually taken with respect to a weight-distribution function ~ ( hwith ) an infinity of points of increase, or more specially a weight-function which is continuous and positive in some interval. T h e principal reference is Szegb’s book; this takes the orthogonality as basic, as do a number of briefer presentations, for example F. G. TRICOMI, “Vorlesungen iiber Orthogonalreihen,” Berlin, 1955,
or D.
“Fourier Series and Orthogonal Polynomials,” Carus Monograph Series No. 6. Ohio, 1941.
JACKSON,
T h e recurrence relation point of view is systematically developed in Stone’s book, pp. 530-614.
Section 4.2 T h e inequality (4.2.4) appears in Sturm-Liouville theory as the monotonic dependence of a certain polar angle on the spectral para-
490
NOTES
meter [Theorem 8.4.3(iii).] Though this is not our approach here, we indicate the very simple proof of (4.2.4) which rests on the orthogonality to be proved in Section 4.4. I t follows from this orthogonality (see Problems 1 and 9) that the polynomial y,JA) ~ Y , _ ~ ( A ) , for any real h, has at least m - 1 changes of sign as A increases on the real axis. But if equality held in (4.2.4) for some real A, this polynomial would, for suitable h, have there a multiple zero, and so would have at most m - 2 changes of sign; the constant sign of the left of (4.2.4) may now be ascertained by considering the highest power of A. For an extension of this argument to higher order Wronskians see the monograph of Karlin and Szego (p, 6), where numerous other interesting investigations will be found. For a converse of the Wronskian property see
+
W. A. AL-SALAM, On a characterization of orthogonality, Math. Mag. 31 (1957/58), 41-44.
Section 4.3 Concerning the zeros of y,(A), as a function of X, see Szegb’s book, Section 3.3, where a variety of arguments is given. T h e oscillatory properties of y,(A), as a function of x, were apparently known to Sturm, though not proved until much later, by M. B. PORTER, On the roots of functions connected by a linear recurrent relation of the second order, Ann. o j Math. (2) 3 (1902), 55-70; see also 0. DUNKL,The alternation of nodes of linearly independent solutions of second-order difference equations, Bull. Amer. Math. SOC.32 (1926), 333-334,
W. M. WHYBURN, On related difference and differential systems, Amer. J . Math. 51 (1929), 265-280.
For a detailed exposition we refer to Fort’s book. See also the book of Gantmaher-Krein, Chapter 2, Section 1, and the monograph of Karlin and SzegB. Anticipating the topic of Chapter 5 in some degree, consider the infinite recurrence sequence defined by C,U,+~ = b,u, - C , - ~ U , - ~ , n = 0, 1, ..., with initial values u - ~ u,, , not both zero. T h e recurrence relation may be said to be “nonoscillatory” if the sequence u, is ultimately of one sign; as in the case of second-order differential equations, this classification is one of the recurrence relation, and does not depend on the choice of initial data. Again as in the case of differential equations, the question has applications to the nature of the spectrum. I n addition to Fort’s book, see for example P. HARTMAN and A. WINTNER, Linear differential and difference equations with monotone solutions, Amer. J. Math. 15 (1953), 131-143,
49 1
NOTES
P. J. MCCARTHY, Note on the oscillation of solutions of second order linear difference equations, Portugal. Math. 18 (1959), 203-205.
T. FORT,“Limits of the characteristic values for certain boundary problems associated with difference equations, J. Math. Phys. 35 (1957), 401-407,
and the notes for Section 5.2.
Section 4.4 The normalization constants pr given in (4.4.34) are essentially the reciprocal of the Christoffel numbers; see Szego’s book, (3.4.7-8), for the case, in our notation, h = 0:
Section 4.5 Our two forms (4.5.4), (4.5.5) conceal a well-known identity in the theory of orthogonal polynomials. Comparing the two we have, taking h = 0, m =
- ~ r n ( h ) J”
-m
(A - PI-’
dTrn.O(p)
sinceym(p)vanishes at the jumps of ~ ~ , , , ( pAnticipating ). the “mechanical quadrature” (5.2.11) [or (4.8.8)] it follows that
This may be interpreted in the sense that we start with the weightdistribution ~ ( h ) construct , polynomials ym(A), by orthogonalization, which necessarily satisfy a recurrence relation, and then derive by the last formula a second solution of the recurrence relation (Szego, “Orthogonal PoIynomials,” Section 3.5). For the continuous analog, relating to solutions of second-order differential equations, see B. M. LEVITAN, On a theorem of H. Weyl, Doklady Akad. Nauk SSSR 82 (1952), 246-249.
Section 4.7 A similar problem has been treated by B. WENDROFF, On orthogonal polynomials, Proc. Amer. Math. SOC.12 (1961), 554-555.
Problems of this kind have interesting mechanical formulations. We refer to the book of Gantmaher-Krein, Appendix 11, “On a remarkable
492
NOTES
problem for a string of pearls and on Stieltjes continued fractions,” where the problem is treated in the form that particles are to be fixed on a light string, with given length and tension and fixed at one end, so as to have one given set of frequencies when the other end is fixed, and another given set of frequencies when this end slides transversely. T h e dynamical interpretation leads to interesting extremal problems, such as minimizing the total mass to be fixed to the string so as to produce given frequencies. See M. G . KREIN,On some problems on the maximum and minimum for characteristic values and on Lyapunov stability zones, Priklad. Mat. i Mekh. 15 (1951), 323-348; or Amer. Math. Soc. Transl. ( 2 ) 1(1955), 163-187, On some new problems of the theory of the oscillation of Sturmian systems, Priklad. Mat. i Mekh. 16 (1952), 555-568,
D. BANKS,Bounds for the eigenvalues of some vibrating systems, Pracific J. Math. 10 (1960), 439-474,
B. SCHWARZ, “On the extrema...,” J . Math. Mech. 10 (1961), 401-422. B. SCHWARZ, Some results on the frequencies of nonhomogeneous rods, J. Math. Anal. Appl. 5 (1962), 169-175,
where references are given to work of P. R. Beesack and S. H. Gould. On the relation to inverse spectral problems see also R. BELLMAN and J. M. RICHARDSON, A note on an inverse problem in mathematical physics, Quart. Appl. Math. 19 (1961), 269-271 ;
references to some analogous problems for differential equations are given in the notes to Section 12.7. In particular, in the Gel’fandLevitan solution of the inverse Sturm-Liouville problem the parallel with the orthogonalization of the powers to form orthogonal polynomials appears to have been found suggestive in connection with the orthogonalization (in a continuous sense) of the function cos Kx.
Chapter 4, Problems Problems 6-10. For the basic theory of CebyBev systems, sometimes called Markov systems when there is an infinite sequence of functions, see Ahiezer’s book, Chapter 2, and the book of Gantmaher-Krein, Chapters 3, 4, where many examples of such systems are found, in association with boundary problems. I t is possible to discuss multiple zeros of linear combinations of functions of such systems, without introducing differentiability. See for example D. R. DICKINSON, On Tschebysheff polynomials, Quart. J. Math. 10 (1939), 277-282; 12 (1941), 184-192; alsoJ. London Math. SOC.17 (1942), 211-217. S.LIPKA,Uber die Anzahl der Nullstellen von T-Polynomen, Monatsh. Math. Phys. 51 (1944), 173-178.
NOTES
493
Problem 15. These are the CebySev-Markov-Stieltjes inequalities. For an analogous property for a second-order differential equation see M. G. KREIN,Analog of the Cebysev-Markov inequalities in a one-dimensional boundary problem, Doklady Akad. Nauk SSSR 89 (1953), 5-8.
Section 5.1 Among illustrations of the theory of this chapter are the classical polynomials of Legendre, Jacobi, Hermite, and Laguerre, discussed in Szego’s book and elsewhere, and certain discrete analogs of the special functions. See R. J. DUFFINand Th. W. SCHMIDT, An extrapolator and scrutator, J. Math. Anal. Appl. 1 (1960), 215-227, P. LESKY, Unendliche orthogonale Matrizen und Laguerresche Matrizen, Monatsh. Math. 63 (1959), 59-83, and the same author’s Die Ubersetzung der klassischen orthogonalen Polynomen in die Differenzenrechnung, ibid. 65 (1961), 1-26; 66 (1962), 431-435.
R. H. BOYER,Discrete Bessel functions, J. Math. Anal. Appl. 2 (1961), 509-624,
and the monographiof Karlin and Szegb. A case when the polynomials have a definite asymptotic form for large n is considered by D. J. DICKINSON, H. 0.POLLAK, and G. H. WANNIER, On a class of polynomials orthogonal over a denumerable set, Paci’cJ. Math. 6 (1956). 239-247.
For other recent work see V. G. TARNOPOL’SKII, The dispersion problem for a difference equation, Doklady Akad. Nauk SSSR 136 (1961), 779-782, W. G. BICKLEY and J. MACNAMEE, Eigenvalues and eigenfunctions of finite difference operators, Proc. Cambridge Phil. SOC.57 (1961), 532-546.
Many special polynomials, some of which have orthogonality properties, have been considered by Carlitz; see for example L. CARLITZ,On some polynomials of Tricomi, Boll. Union Mat. 1201. (3) 13 (1958), 58-64.
Section 5.2 T h e observation that a three-term recurrence relation of a suitable form defines polynomials which are necessarily orthogonal on the real axis seems to have been first explicitly stated by J. FAVARD, Sur les polynomes de Tchebicheff, C . R. Acad. Sci. 200 (1939, 2052-2053,
who remarked that the result followed from one of Hamburger.
494
NOTES
It would appear that the result was already in the possession of J. SHOHAT, The relation of the classical orthogonal polynomials to the polynomials of Appell, Amer. J. Math. 58 (1936), 453-464.
For related earlier investigations see E. HELLINGER, Zur Stieltjesschen Kettenbruchtheorie, Muth. Ann. 86 (1922), 18-29,
J. SHERMAN, On the numerators of the convergents of the Stieltjes continued fractions, Truns. Amer. Muth. SOC.35 (1933), 64-87,
and the later sections of Stone’s book. It is also possible to consider recurrence relations in which our restrictions of sign on the coefficients are relaxed in an essential way, and orthogonality with respect to a distribution of bounded variation, which need not be nondecreasing. See J. SHOHAT, Sur les polynomes orthogonaux gknkraliskes, C . R . Acad. Sci. 207 (1938), 556-558,
D . DICKINSON, On certain polynomials associated with orthogonal polynomials, Boll. Union. Mat. Iltal. (3) 13 (1958), 116-124.
In forming the sequence of spectral functions ~ , , , ~ ( hit )is permissible to restrict ourselves to the case h = 0, so long as we merely wish to show that there is at least one limiting spectral function. In this case the spectral functions have their jumps at the zeros of ym(h),rn = 1, 2, ..., and this leads to the conclusion that the interval of orthogonality may be taken to be the smallest interval containing all the zeros of all the yl(h). This interval is sometimes termed the “true” interval of orthogonality of the polynomials. A particularly important case is that in which the zeros of the y , ( h ) have one sign only. This occurs in the case of the vibrating string and in the case of recurrence relations associated with birth and death processes. Concerning the latter, see for instance S . KARLIN and J. MCGREGOR, Linear growth, birth and death processes, J . Math. Mech. 7 (1 958), 643-662.
T h e situation in which all the polynomials have zeros of the same sign has recently been studied by T. S . CHIHARA, Chain sequences and orthogonal polynomials, Trans. Amer. Math. SOC. 104 (1962), 1-16.
The case of quasi-orthogonal polynomials, orthogonal when their degrees differ by at least two, has been considered in regard to necessary and sufficient conditions by D. DICKINSON, On quasi-orthogonal polynomials, Proc. Amer. Math. SOC.12 (1961), 185-194.
495
NOTES
Section 5.4 The nesting circle aspect of the second-order difference equation was brought out by E. Hellinger [Math. Ann. 86 (1922), 18-29], in analogy to the famous discovery of H. WEYL,Uber gewohnliche Differentialgleichungen mit Singularitaten und die zugehorigen Entwicklungen willkiirlicher Funktionen, Math. Ann. 68 (1 910), 220-269.
The argument leading to the nesting property can be extended to allow the b, to have suitably restricted complex values; furthermore, the X appearing in the recurrence relation may depend on n, the copplex X with Im X > 0 being replaced-by a sequence of values in the bpper half-plane. See Wall’s book, Chapter 4, and the work of Sims referred to in the notes for Section 8.13.
Section 5.7 In the limit-circle case, the existence of a plurality of spectral functions, and of orthogonality relations, both direct and dual, leads to a plujality to solutions of certain differential equations, similar to (0.7.16), or to a plurality of values of the exponential of a certain matrix, similar to (0.7.15). The situation is of interest in connection with birth and death processes, and for a fuller discussion we refer to the article of W. FELLER, The birth and death processes as diffusion processes, J. Math. Pures AppI. 38 (1959), 301-345,
where other references are given. We vary our notation, changing the sign of X and replacing an by I, so that the polynomials are to be defined by
+ +
~n~n+l(h)
+
bn) ~ n ( x )
cn-~n-l(A)
= 0,
with c, > 0, yPl(A)= 0, C-~~,,(X) = 1. We shall assume that the spectrum is bounded from below, and that the limit-circle case holds. In other terms, for any solution of the recurrence relation with X = 0, that is any sequence u, satisfying cnu%+l
+
b%un
+
cn-lUn-1
= 0,
the terms must be ultimately of one sign, the equation being “nonoscillatory,” and the sequence must be of summable square, or Z: I un l2 < 00. For any chosen real A’ we may then determine a boundary problem by (5.7.4), with a direct orthogonality
496
NOTES
the dual orthogonality (5.2.4) holding with a, = 1; here ~ ( his) a step function with jump l/p, at A, , and will also be a limiting spectral function. We now set up the expressions, for t >, 0,
p ( j , k, t ) =
J
m
-m
e - l t y j ( ~Yk(h) ) dT(h),
which have many interesting properties. In the first place we have immediately that p ( j , k, 0 ) = 6 j k . Furthermore, by the recurrence formulas for the y,(A), (d/dt)p(j,k, t ) = cip(j = ckp(j,
+ 1, k,t ) + M j , k, t ) + ~ i - ~ p-( j 1, h, t ) - 1, t ) . k + 1, t , f k, t ) + bkp(j,
ck-lp(j>
From these, and the facts thatp( - 1, k, t ) = p ( j , - 1, t ) = 0, p ( j , j , t ) >O, it may be verified that p( j , K, t ) > 0 for t > 0. We mention in passing that any spectral function .(A) gives us a solution of the above system of differential equations with the same initial conditions. That these solutions are in fact different, for different .(A), may be seen by considering the asymptotic nature of p(0, 0, t ) as t --t 00. In fact, we need not confine attention to limiting, or orthogonal, spectral functions. T h e breakdown in the uniqueness theorem for differential equations with given initial data is due, in part, to the fact that we have a differential equation with an infinity of unknowns. Confining ourselves to spectral functions arising from a boundary condition of the above type, so that the eigenfunctions are orthogonal, we assert the “semigroup” property, that there hold the “ChapmanKolmogorov equations” m
for s >/ 0, t >, 0. This is immediately to be verified, on writing out the p’s as sums and using the orthogonality of the eigenfunctions. Again, we have an infinity of solutions of these relations, but with a more restricted class of spectral functions. With this semigroup property and the non-negativity of the p ( J , k, t ) for t >, 0 we approach the conditions defining a Markov process. Moving further in this direction, and without confining ourselves to the limit-circle case, let us assume that b, co = 0, b, c1 c,, = 0, ...; it may then be verified that Zk( d / d t ) p ( j ,k, t ) vanishes, so that & p( j , K, t ) is constant, and so unity, the same conclusion holding for & p ( j , k, t ) . T h e infinite matrix p( j , k, t ) is thus “doubly stochastic” (Bellman, “Matrix Analysis,’’ pp. 267-268).
+
+ +
497
NOTES
A further possible development in the limit-circle case is to impose a boundary condition at infinity containing a parameter. This is readily interpreted in the case of a vibrating string of finite length, bearing an infinity of particles of finite total mass converging to one end, a finite particle being located at that end, and free to slide transversely. For such a development in the probabilistic context we refer once more to Feller’s paper. Concerning that fact the the integrals defining the p( j , k, t ) are not independent of the choice of ~ ( h )in, the limit-circle case, we reach an apparent contradiction with this fact on expanding the exponential exp ( - A t ) in a power series and integrating term by term; the resulting integrals of polynomials in h should be independent of the choice of ~ ( h )T. h e resolution of the difficulty is that in the limit-circle case ~ ( h ) does not tend to its limits as h -+ 3 00 sufficiently rapidly to justify the term-by-term integration. See problem 11 for this chapter, or Titchmarsh’ “Fourier Integrals,” p. 320. These investigations for polynomials with scalar coefficients would seem to admit extension to the case of polynomials with matrix coefficients. See Sections 6.6-8 of this book and, for the probabilistic aspect, R. BELLMAN,On a generalization of classical probability theory. I, Markoff chains, Proc. Nut. Acad. Sci: U.S.A. 39 (1953), 1075-1077.
Another treatment of the subject has been given by J. H. B. KEMPERMAN, An analytical approach to the differential equations of the birth and death processes,MichiganMath.J. 9 (1962), 321-361.
Regarding the convergence of the formal series expansion of the integrals defining p ( j , k, t ) and the moment problem, see R. FORTET, Calcul des moments d’une fonction de rtpartition B partir de sa caracttristique, Bull. Sci. Math. 68 (1944), 117-131.
Section 5.8 T h e distinction between limit-point and limit-circle cases may be carried out in a functional-analytic context. T o simplify matters we suppose that a, = 1 in (5.1.1); this may be achieved by a substitution yk = a;l2y,, bk = a;’J2b,, c i = a;’J2c,a;Y: . This done, we consider the Hilbert space l2 of sequences of complex numbers such that
5
= (50
1 5 ,
9
...I
498
NOTES
the scalar product being given by
T h e transformation 5 -+ C’, where the components of in terms of those of 5 by
5;
= Cnln+l - bn5n
+
5’
are given
Cn-,5n-l>
where formally we set = 0 or c - ~= 0 for the case t z = 0, then defines a linear operator within 12, which will be denoted differently according to the domain, that is to say, the subset of l2 to which the transformation is applied. As a minimal domain of definition of this transformation let us take the set lo formed by sequences 5 such that only a finite number of the 5, are different from zero. We denote by A the linear operator given by 5’ = At for 5 E I , , so that I, is the domain D, of A. It is then easily verified that, if 5, r] E D,, then in other words, that
(47) = (5, AT),
where the 5, , 71, are all zero beyond some point. This means that A is symmetric, or Hermitean. However, A is not self-adjoint. T o define the adjoint A* of A we consider the set of 7 E l2 for which there is an 7’ E l2 such that
(A597) = (597’) for all 5 E D, = lo; the set of such 9 forms the domain of A* and on it we have r]’ = A*r]. As we have seen, if r] E lo we may take r]‘ = Aq. However, it is not hard to show that r]’ exists also for some r ] E l2 not in 1,; it i s sufficient to require that
This means that A* agrees with A on l o , but is also defined on a larger set, so that A* is an extension of A; since A* does not coincide with A , the latter is not self-adjoint.
499
NOTES
Consider next the operator B defined in the same way, that is, by = B5 with 5’ as above, but with the maximal domain of definition as an operator on 12 into 1 2 ; the domain D , consists of the set of 5 E l2 such that
5‘
It may happen that B is self-adjoint; this is in fact the limit-point case. T h e simplest case is that in which the constants b , , cn are bounded, uniformly in n. Here the operator B is bounded, and its domain is the whole of 12. T h e condition for self-adjointness coincides with that for symmetry, in this case that
(a, 7) = (1, B7)l for all
5, n E 12.
-
-
On calculation we find that this is equivalent to Cn(LL+,;in
- Slliin+l)
0
as
0°*
This i s true since we assume c, bounded and since 1,-0,
vn-0
1 , EP. ~
for
It is easily seen that the case in which B is bounded belongs to the limit-point case. Supposing the limit-circle case to hold, and writing y(A) for the sequence formed by {y,(A),y,(A), ...}, we should have y(A) E 12, by (5.4.7), and also By(A) = Ay(A), by (5.1.1-3). This is impossible for large A, and so the limit-point case holds. If we merely assume the c, uniformly bounded, the domain D, is characterized by Z I b,, 0 at a local minimum off. For exact statements see W. FELLER, On the intrinsic form for second order differential operators, ZllinoisJ. Math. 2 (1958), 1-18, Differential operators with the positive maximum property, ibid. 3 (1959), 182-186.
NOTES
515
A third aspect is the relation of these two general properties, enjoyed in particular by the mixed derivative, to general categories of diffusion processes. See W. FELLER, The general diffusion operator and positivity preserving semi-groups in one dimension, Ann. of Math. 60 (1954), 417-436.
Some differential equation cases are considered by P. MANDL,Spectral theory of semi-groups connected with diffusion processes and its application, Czech. J. Math. 4 (1961), 559-569.
Another school of investigation, using the formalism of integral equations (see Chapters 11 and 12), is associated with M. G. Krein, another contributor being I. S. Kac. Of Krein’s many papers we cite particularly On a generalization of investigations of Stieltjes, Doklady Akud. Nauk SSSR 87 (1952), 881-884.
Aspects relating to the spectrum are developed in I. S. KAC,On the existence of spectral functions of certain singular differential systems of the second order, Doklady Akad. Nuuk SSSR 106 (1956), 15-18, On the behavior of spectral functions of differential systems of the second order, ibid. 183-186,
more references being given in the same. author’s Growth of spectral functions of differential systems of the second order, Izw. Akad. Nauk SSSR, Ser. Mat. 23 (1959), 257-214.
I n addition to Volterra integral equations, it is also possible to consider Fredholm integral equations with Stieltjes weight distributions as generalizing Sturm-Liouville theory. This approach is developed in the book of Gantmaher and Krein. See also M. G. KRE~N, On the Sturm-Liouville problem in the interval (0, a)and on a class of integral equations, Dokludy Akud. Nuuk SSSR 7 3 (1950), 1125-1 128
or, for a simple illustration, Bellman’s “Matrix Analysis,” p. 144, Exercises 1 and 2. For the presentation of Sz.-Nagy we refer to B. SZ~KEFALVI-NACY, Vibrations d’une corde non homogene, Bull. SOC.Math. France 75 (1947). 193-208,
or to the book of Riesz and Sz.-Nagy, where the spectral resolution is derived by methods of functional analysis. We refer under the notes for Sections 0.8 and 8.7 to investigations of the slightly more special situations of classical Sturm-Liouville theory
516
NOTES
modified by the presence of the parameter in the boundary conditions, or by a finite number of interface conditions; under the notes for Section 11.8 we refer to work on more general systems of higher dimensionality. A number of new directions have been opened up in Sturm-Liouville theory under classical continuity conditions. A survey is given by B. M. LEVITAN and I. S . SARGSYAN, Some problems in the theory of the Sturm-Liouville equation, Uspekhi Mat. Nauk 15, No. l(91) (1960), 3-98.
That recurrence relations may be imbedded in the theory of differential equations by taking the coefficients to be piecewise constant is explained in a matrix context by W. T. REID,Generalized linear differential systems, J . Math. Mech. 8 (1959), 705-726 (in particular pp. 721-722),
where reference is made to the dissertation of V. C. Harris.
Section 8.3 Concerning the e3ponent of convergence of the zeros of an entire function, relevant to the proof of (8.3.7), see for example Titchmarsh, “Theory of Functions,” Section 8.22. For more special situations than those considered here, we may obtain more information on the distribution of the eigenvalues either by classical methods (as in the text of Ince) or from more incisive results from the theory of functions; see for example the result of Levinson discussed in P. Koosrs, Nouvelle demonstration d’un theoreme de Levinson ..., Bull. SOC.Math. France 86 (1958), 27-40.
For the asymptotic form of the eigenvalues of a vibrating string with arbitrary mass-distribution see M. G. KRE~N, Determination of the density of a symmetrical inhomogeneous string from its spectrum of frequencies, Doklady Akad. Nauk SSSR 76 (1951), 345-348, and On inverse problems for an inhomogeneous string, ibid. 82 (1952), 669-672.
For cases in which the eigenvalues increase more rapidly than the classical estimate O(n2),see H. P. MCKEANand D. B. RAY, Spectral distribution of a differential operator, Duke Math. J . 29 (1962), 281-292.
Since the eigenvalues are the zeros of the left of (8.3.4), an entire function of order less than 1, we may by factorizing this function
517
NOTES
[cf. (12.3.27)] obtain explicit formulas for the sums of inverse powers of the eigenvalues. See R. BELLMAN, Characteristic values of Sturm-Liouville problems, Illinois 3.Math. 2 (1958), 577-585.
For trace-formulas involving sums of eigenvalues see L. A. D I K I ~Trace , formulas for Sturm-Liouville differential operators, Uspekhi Mat. Nauk 13, No. 3(81) (1958), 111-143.
Section 8.4 T h e use of the polar coordinate method to establish the Sturmian oscillatory properties seems due to H.Prufer (see the notes for Section 8.6). T h e key fact that the polar angle, as defined in this particular version of the method, is a monotonic function of the spectral parameter was also noticed by W. M. WHYBURN, Existence and oscillation theorems for non-linear differential systems of the second order, Trans. Amer. Math. SOC.30 (1928), 848-854 (p. 854), and A non-linear boundary value problem for second order differential systems, Pacific Math. 5 (1955), 147-160,
3.
where nonlinear systems are also treated. A second and distinct version of the polar coordinate method belongs in the area of asymptotic theory, either for large parameter values or for large values of the independent variable; we have used this method, in a somewhat crude form, at the end of this section. A similar device is used at the end of Section 10.5. In its more precise form, this other version of the polar coordinate method applies to the second-order equation y f f f ( x ) y = 0, where f ( x ) is smooth and positive, and may depend on a spectral parameter. T h e method depends on an investigation of the differential equation of the first order satisfied by O(x) as defined by tan B = -y’/(yf1l2); the success of the method depends, roughly speaking, on the variation in logf being small compared to the integral of fl2. For applications of this method see
+
F. V. ATKINSON, On second-order linear oscillators, Rev. Univ. Tucuman, Ser. A , Mat. y Ffs. Tedr. 8 (1951), 71-87,
J. H. BARRETT, Behavior of solutions of second order self-adjoint differential equations, Proc. Amer. Math. SOC.6 (1955), 247-251,
J. B. MCLEOD,On certain integral formulae, Proc. London Math. SOC.(3) 11 (1961), 134-138, and
518
NOTES
The distribution of the eigenvalues for the hydrogen atom and similar cases, ibid. 139- 158,
H. HOCHSTADT, Asymptotic estimates for the Sturm-Liouville spectrum, Comm. Pure Appl. Math. 14 (1961), 749-764, N. WAX, On a phase method for treating Sturm-Liouville equations and problems, J. SOC.Ind. Appl. Math. 9 (1961), 215-232, N. S. ROSENFELD, The eigenvalues of a class of singular differential operators, Comm. Pure Appl. Math. 13 (1960), 395-405.
For ramifications of the method reaching into optics, statistical mechanics and quantum theory, particularly the so-called WKB method, see P. FRANKand R. VON MISES,“Die Differential- und Integralgleichungen der Physik,” Vol. 11. Braunschweig, 1935, pp. 82, 119, 986.
+ +
For equations of the form y” (k2 g(x)) y = 0, where g(x) need not be smooth but is in some sense small, one may modify the substitution to tan 8 = -y’/(ky); see Problems 5-7 for Chapter 12. Returning to separation, comparison and oscillation theorems, we refer to books such as that of Ince for a treatment of these topics not involving the polar coordinate method; in this work nonlinear dependence on the parameter is also considered, with a view to the multiparameter application. For results for more general types of side condition see W. M. WHYBURN, Second-order differential systems with integral and k-point boundary conditions, Trans. Amer. Math. SOC.30 (1928), 630-640.
Section 8.5 Regarding the Cebygev property in general see the Notes to Chapter 4, Problems 6-10, and the books of Ahiezer and Gantmaher-Krein. That the eigenfunctions of a Sturm-Liouville problem have this property, in its fullest statement that a linear combination of u,(x), ..., us(x), 0 0 are real, then the eigenvalues A, for which r > 0 satisfy
Show also that this bound can be attained, for suitable c, 4. Show that if the c,, > 0 are real and a = m, then
.
5. Shown that if Rl {c,} > 0, and m < m’ , then between two eigenvalues h,(m, a),&+l(m, a) lies an eigenvalue h,(m’, /3). 6. Consider the boundary problem
+
( E ihcn) ( E - ihC,*)-’yn YO # 0 Ym = NY, where yn is a k-by-1 column-matrix, E the unit matrix, N a fixed unitary matrix, and the matrices C, are such that C, C,* > 0, C,C,* = C,*C, showing in particular that the eigenvalues are real. 7. For the problem of the previous question, and supposing the C, Hermitean, establish the analog of Problem 2, for the event that one of the C, is increased in the matrix sense. (Remark: Suitable arguments are used in a different context in Chapter lo.) ~ n + l=
9
+
536
537
PROBLEMS
= 0, ..., m - 1, Y = I , ..., k let the , ..., Cn7k) be real 1-by-k row matrices, and let = (A,, ..., A,)
8. For a fixed integer k > 1 and n c,
= (c,,,,
be a k-by-1 column matrix. Define the y,,,(h) recursively by y,(X)
+
= 1 and
~ n + l , d X )=z (1 icnJ) (1 - iCn4)-'Ynr(A)* For some real a,, ..., ak form the boundary problem
ymr(h)= exp (ia,), r = 1, ..., k, eigenvalues being column matrices X which satisfy these k simultaneous equations. For a k-tuple n of integers (n, , ..., nk) , 0 n, < m , write
Ofor all n, then this boundary problem has only real eigenvalues. Prove also that it has exactly mk eigenvalues, and set up orthogonality relations. (Remark: Compare Sections 6.9-10, Problem 16 for Chapter 8). for the determinant formed by the k row matrices cni1,
Chapter 2. 1. Show that the eigenvalues of the problem (2.5.1-3) with (Y = &r, c,, = n-2, are 12/2, 32/2, ... and - 2.12, - 2.22, ... . 2. Show that if the c,, are real and positive and such that c,, = O ( r 2 ) for large n, then A'; = O(Y-~)for large Y. 3. For the matrix analog of the problem (2.5.1-3), i.e., Problem 6 of Chapter 1 with m = 00, prove that the spectrum is discrete provided that m
0
4. Consider the multi-parameter boundary problem described in Problem 8 of Chapter 1 with m = m, the c, being real and all c, > 0. Show that if
exists for all real h and for suitably restricted complex A. Show also that the spectrum is discrete, i.e., that the eigenvalues (which are column matrices) have no finite limit. Chapter 3.
+
1. Let A, B be 2-by-2 matrices and let A XB for all real X be symplectic AB = A'(] AC), [i.e., ]-unitary with ] given by (3.5.1)]. Show that A
+
+
538
PROBLEMS
where A' is symplectic, and independent of A, and C has one of the forms
3
and a, b are real. 2. In addition to the assumptions of Problem 1, let Im ( ( A
+ hB)*/(A + m) - /} 2 0,
for all h with Im h > 0, with equality excluded in the matrix inequality. Show that in the above form for C the (+) sign is to be taken, with a and b not both zero. 3. Let A,, AB,, satisfy the assumptions of Problems 1 and 2. Show that the recurrence relation, for the column matrices yn ,
+
yn+1
=
(An
+W y n,
71
= 0, 1,
.** >
may be transformed, by a substitution z,, = H,,y,, with H,, symplectic, to the form n == 0,1, ..., Zn+l = (/ XC,,)Z,, ,
+
where C,, has the form attributed to C in Problems 1 and 2. 4. With the assumptions of the previous problems, show that a further substitution of the form
reduces the recurrence relation to the form
and so to a scalar three-term recurrence formula. (Hint: Two recurrence steps of Problem 3 for which C,,+,C,, = 0 may be combined into a single step, and the sign of a,,, b, may be adjusted so that anan-, b,,b,,-, > 0.) 5. Let A , B be 2-by-2 matrices and let A +hB be unitary, in the usual sense, for all h such that 1 h I = 1. If further A , B are neither equal to the zero matrix. show that
+
where A', A" are unitary and independent of A. 6 . Show that a recurrence relation Y,,+~ = (A,, AB,,)y,,, where A,,, B,,
+
are as in the previous problem, may be transformed by a substitution z,, = Hnyn to the form 0 cos B,, sin /I,,
539
PROBLEMS
(Equivalently, the order of the matrices may here be ,reversed.) 7. Let A , B be 2-by-2 matrices, neither zero, such that for all A on the unit circle A AB is J-unitary, where 1has the form (3.6,l).Show that A hB has one or other of the forms
+
+
where A‘, A” are J-unitary. 8. Find a parametric representation of the general J-unitary matrix, with J given by (3.6.1),and a standard form similar to that of Problem 6 for the associated recurrence relation.
9. Show that the set of 2-by-2 symplectic matrices
(,“ ):
is connected when
u, b, c, d may be complex, and also when they are restricted to be real.
Chapter 4. 1. Polynomials y,,(A) are defined as in Section 4.1, and the constants a,. , m, Y m2 , 0 < m, m2 m, are not all zero. Prove that XZ2 ary,.(A) 1 has at least m, , and at most m2 real zeros. (Hint: This may be deduced from the orthogonality.) 2. The piecewise linear function yz(A) is defined as in Section 4.3,and h, , A,,+l are the zeros of ym(A). Show that a linear combination
<
k a polynomial y,,(h) can have at most k 1 zeros in h >, A”. (Hint: Use Theorem 4.3.5.) 6. Let (b, - c,, - c,,-~)/u,, -+ m as n -+ 03. Show that there is a discrete spectrum, any limiting spectral function being a pure step function. 7. If the assumptions of both Problems 1 and 2 hold, show that the spectrum is finite and the limit-point case holds.
+
+
+
+
8. Show that, for the recurrence relation for Hermite polynomials, the limitpoint case holds. p e have here a,, = 1/(2*-1n!), b,, = 0, c, = 1/(2”n!), and it is known that HZ,(O) = (- 1),(2n)!/n! .] 3. Let polynomials $,,(A) be defined by P - ~ ( X ) = 0, p,,(h) = 1, and
%A+l(4= (an
+ AI - 4Pn(h) - 15,p,-l(h)
(an
, F,, > 0).
Show that the polynomials are orthogonal with respect to a weight distribution located in 0 X < m.
n log n + Bn
for n >, 1 and some constant B.
[Hint: Carleman’s theorem (Titchmarsh, “Theory of Functions,” Section
3.7) provides one method.]
544
PROBLEMS
Chapter 6. 1. Show that the Green's function g&) representation g7m =
2
defined in Section 6.4 admits a
YA~P)Ys(AP)P;'(~P
-
P
4-l
'
where the A, are the zeros of ym(A)and p D is given by (4.4.34),with h = 0. 2. Show also that g,,(h) may be specified as a rational function which tends to zero as h -+ 00, and which differs from z,(X)yl.(X)y,(h)/r,(h) by a polynomial. 3. Let the matrix polynomials Yn(h),Y:(h), Zn(h), Z$h) be defined by the recurrence relations
Yn+l = ( A J
+ Bn)Yn- Yn-l
YJ+l= Y:(A,X Zn+1 = ~ : + 1=
+ B,)
Y-1 = 0,
Yo = E ,
YTl
= 0,
YJ = E,
Zn-1,
2-1
= E,
Zo
Z nt - 1 ,
Z!,
= E,
2 : = 0,
- Yi-,
+ B,)Zn Z,t(AnX + Bn) (Anh
,
= 0,
capital letters denoting K-by4 matrices and 0 and E the zero and unit matrices. Show that --E
0
Yn+l
Yn
',+I)
zn
=
(i
4. Show also that, with the same notation as in Problem 3,
5. Show further that, apart from poles,
Y;l(h)Zn(h)= -
n-1
Y;;l(h)Y:-l(A) .
0
6. Show that, for polynomials R(X),S(X), whose coefficients are square matrices, the integral
I
R(h)YG1(A)Zn(X)S(h) dh ,
caken round a closed contour in the positive sense, the contour enclosing all poles of Y;l(h), has the same value for all sufficiently large n, namely, such that 2n exceeds the total of the degrees of the fixed polynomials R ( 4 , S(4.
545
PROBLEMS
7. Let the k-by-1 column matrix y,,, - 1 < n = 0, for a square matrix H , and ym + yn+1
- (AT?
+ BnlY,, +
yn-1
< m, satisfy
(n # O),
= 0,
= u,
y-l
=
0,
(n = O),
for a column matrix u. Show that, apart from poles,
yo
=
+ Hym-l(4}-1{zm(4 + HZ,-1(4}u
{Ym(h)
8. Assume A,,, B,, , H Hermitean, A, - {ym(h)
*
> 0. Show that
+ H Y ~ - ~ ( A ) } - ~ { +z ~~(zAm) - l ( h ) ) / =
m -m
dTrn,H(p)
- p)-l
where ~ , , ~ ( pis) the matrix-valued spectral function defined in (6.8.9).
9. Let R(X), S(h) be polynomials with matrix coefficients, of total degree less than 2m. With the assumptions of the previous problem, establish the “mechanical quadrature,” according to which the integral
has a value independent of M and of H . 10. With the same restrictions on A,, , B,, , and H show that
+
-{ym(~>
J
~ y m - 1 ( ~ ) } -1 1
m
drrn,H(p)Ym*-l(p)( A - p)-1.
--OD
1 1. Let A,, , B,, be Hermitean, A,, >0, and let H satisfy Im H =(2i)-l ( H -H * ) > 0. Show that the eigenvalues of (6.6.5-6) lie in the lower half-plane. 12. With the assumptions of the previous problem, and with R(h), S(A) matrix polynomials of total degree not exceeding 2m - 2, show that (suppressing the A’s in the integral)
m
= rr
+
-m
R dTrn,os .
+
13. Writing Fn,H = - {Y,, HY,,-l}-l{Z,, HZ,,-l}, show that Fn+l,H(X) = F,~p(h),where H = - (&A B, H)-l. 14. Denote by gm(X) the set of Fm,H(h)for fixed h with Im h > 0, and all possible H with Im H > 0, the A,, B,, being Hermitean with A,, > 0 . Show that, for m 2 1, g m ( h ) is a finite region, and that .9m+l(h) C gm(A). 15. Let ~ ( h ), 00 h 00, be a k-by-k Hermitean and nondecreasing matrix function, such that, for all n 3 0,
+ +
<
0.1 Show that matrix polynomials of the form P,(h) = h"E
c)
>
+ 2 Cn$ n-1
r=o
satisfying the conditions
jrn P,(h) dT(h)hs -w
0,
S = 0,
..., 71 - 1,
exist and are unique. 16. Show that these polynomials minimize the expression
j-
P,,(X) d+) P,*(h), tr -m for varying C,,, . 17. Show that the PJh) satisfy a three-term recurrence relation. 18. Evaluate the polynomials of Section 6.9 explicitly in the case that c,,, = 1, b,,, = 2, and the anrsare independent of n; verify the oscillation theorem for this special case.
Chapter 7.
1. With the assumptions of Section 7.1, if 0 < a < solutions of um(h) - e%,(X) %,(A) - ei%,(h) = 0,
< 27r
show that the
=0
separate one another on the unit circle, in that as h moves positively between two roots of one, it passes through a root of the other. 2. With the same assumptions, and 0 < n < m, show that the solutions of .,(A) - e%,,(h) = 0 are separated by those of urn@)- &%,(A) = 0. 3. With the same assumptions, and I p I < 1, prove that the zeros of
for varying A, lie outside the unit circle. 4. Prove that the zeros of .,(A) lie inside the unit circle, as a consequence of the orthogonality (7.3.1 1). [Hint: In the contrary event, a. polynomial of degree less than n and not orthogonal to .,(A) could be constructed.] 5. Discuss whether the u,(h), .,(A) are determined, or determined apart from constant factors, by a knowledge of the eigenvalues of the two boundary problems of Problem 1, and whether the resulting recurrence relations satisfy the assumptions of Section 7.1.
547
PROBLEMS
6 . Prove that the zeros of un(A) - hu,-,(A), where h is any constant, lie in the circle 1 A l2 < 2, with at most one exception.
The following examples relate to the situation in which the orthogonality has been transferred to the real axis. It is to be understood that the weight distributions on the real axis satisfy such bounds as ensure the absolute convergence of the integrals which occur. Orthogonality is in the complex sense, in that u(A) is orthogonal to v(A) if the integral of u(A)vKwith respect to the weight distribution in question, vanishes. Unless otherwise indicated, the weight-distribution functions .(A), .(A) will be real-valued and nondecreasing. 7. If .(A) is not a constant, and W , , w 2 are complex, with imaginary parts of opposite signs, show that
J
m -w
Jm-m
(A - w,)-l(A
-
du(A) # 0,
wz)-1
(A - W1) (A - W * ) &(A)
# 0.
w < 0 and .(A) has at least n points of increase, show that there is a unique polynomial pn(A), of precise degree n and with unit coefficient of An, which is orthogonal, with respect to .(A), to all polynomials q(A) of degree a t most n which vanish when A = w. 9. Show that the polynomial pn(A) just specified has all its zeros in Im A > 0. 10. The numbers vr , r = 0, ..., m are complex, all distinct, snd such that Im vr < 0. The nondecreasing function T(A) has at least m points of increase. Show that, for 0 p < m, a linear combination ?(A) of ( A - vs)-lI 0 s p , which is orthogonal to these functions, in the sense that
8. If Im
0, by rl > 0, y = y1 sin 8, y' = k ~ ,cos 8. Writing 8 k for a8/ak, show that 8k =
2{r1(x, k)}-2
jz{y(t,k))2 dt + k-' 0
sin 6 cos 8.
10. With the assumptions of the previous two problems, for b > 0 and sufficiently large positive integral n let k,(b) be the value of k for which 8(b, k ) = n ~ so, that A, = k: is that eigenvaiue of
563
PROBLEMS
such that y has tl - 1 zeros in 0 < x < b. Show that as b -+ + the A,,@) become arbitrarily dense in any interval on the real and positive 00,
k-axis. If in particular b -+ 00, and n a positive limit k, show that
-+
00
in such a way that k,(b) tends to
I”
{k,+l(b) - k ? m } where r,(k)
=
lim
yl(x,
Z+W
0
dt
{r
T*
-+
[Note: In the notation of (12.7.6), r,(k) = k-lr(k).] 11. For the boundary problem of the previous problem and finite 6 > 0, let T b ( h ) be the spectral function, so that for A > 0,
Show that, for fixed A‘, A”, with 0
< A‘ < A”,
as b - m . 12. Let a(x), 0 < x < T , be real-valued, right-continuous, and of bounded variation, and let U(X) = u(x, A), W(X) = w(x, k) be solutions of
[Y‘I
+ J”Y d{k2x + +)}
=0
with the initial or final conditions u(0) = 0, u’(0) = 1 and w’(T) = 1. If u ( x ) # 0, define the Green’s function
w(w) = 0,
Show that this function provides, in the usual manner, the solution of the inhomogeneous boundary problem [z’]
+ J” x d{k2x + +)}
=
J” ~ ( x dx, )
z(0) = Z ( T ) = 0, for arbitrary continuous ~ ( x ) . 13. For the problem [Y‘I
+ I Y d{”. + +)}
= 0,
Y(0) = Y ( 4 = 0, let A, , n = 1, 2, ..., be the eigenvalues and y,,(x) the normalized eigenfunctions [(12.9.4) with b = T ] . Show that the Green’s function G(x, t , A) defined in the previous problem has the following properties:
564
PROBLEMS
(i) at an eigenvalue An the Green’s function has a simple pole with residue Yn(tlYri(x> (ii) if h is not an eigenvalue, the Fourier coefficient of G(x, t , A) with respect to r,&)is m(.)/(h - hn), (iii) if h is not an eigenvalue, 3
(iv) if h is complex, G(x, t , A) - G(x, t , h) = (h - A)
I”
G(x, s, h)G(s, t , A) ds,
0
(v) G(x, x, A) is a “negative imaginary” function (see Appendix 11), (vi) if Im h > 0,
o < 2 I yn(x) 12 Im 1/(hn - A) < - I m ~ ( x x,, A). W
1
14. With the notation of the previous problem, show that
I m ( x ) l2 Im I/&
- A),
- Im G(x, x, A)
tend to zero on a contour given by X = k2,where k describes the rectangle R1 k = 0, (m Im k = 0, (m &)r, and m +03. 15. Deduce, from the maximum principle for harmonic functions, that these two functions are equal. Deduce that the bilinear expansion of the Green’s function holds in mean-square, and deduce further the eigenfunction expansion.
+ a)~,
+
Index
Abdel-Messih, M. A. 501 Adern, J. 482 Agranovit, Z. S. 532 Ahiezer, N. I. (Achieser) 478, 484-5, 492, 510, 513,522-3 Al-Salarn, W. A. 490 Ambarzurnian, V. 531 Analogies between difference and differential equations 1, 30, 487, 491-3, 529-530 Andersen, E. S. 506 Aronszajn, N. 486, 533-4 Artin, E. 489 Asymptotic behaviour, of eigenfunctions and, eigenvalues for integral equations 398-41 I , also qus. 6-7 for Chapter 12. of eigenvalues for differential equations 208, 319-323, 338, 516 of phase 532, qu. 7 for Ch. 8, qu. 5 for Ch. 12. of polynomials orthogonal on real interval 196-8 of polynomials orthogonal on unit circle 190-6, 508, 510 of solutions of differential equations 206-7, 366, 449-452, 517 of solutions of integral equations 366370, 384-398 of spectral function 56, 61, 130, qu. 9 for Ch. 8, qu. 1 1 for Ch. 12. Atkinson, F. V. 504, 506, 517, 529 Balogh, T. 511 Banks, D. 492 Bari, N. K. 535 Barrett, J. H. 517, 523, 525-6 Barrett, L. C. 520
Bauer, W. F. 482 Baxter, G. 504-8, 510, 512 Bazanov, B. V. 500 Beckenbach, E. F. 478, 487, 500, 520, 534 Beesack, P. R. 492 Bellrnan, R. 377, 453, 455, 451, 476, 478, 481, 492, 496, 501-3, 515, 520, 524, 528, 531, 534. Bickley, W. G. 493 Bieberbach, L. 478, 513, 520 Birkhofl, Garrett, 475, 478,513, 533-535 Birkhoff, George D. 478,521, 523,533 Birth and death process 19, 494-5,497 Rlaschke product 58,487 Bliss, G. A. 521, 523, 525 Bochner, S. 518 Borg, G. 531 Rott, R. 523 Boundary conditions, at infinity, 64, 120, 132-3, 497, 499, 501 general form of, 14, 84, 88, 89, 255-6, involving parameter, 22-4, 482-3, 5 16, 520 involving second derivative 482 periodic 93, 145, 256, 336 Boyer, R. H. 493 Brauer, F. 522 Brodskii, M. S. 485, 532 Bureau, F. J. 484 Cargo, G. T. 487 Carlitz, L. 493 Carrnichael, R. D. 503 Cebyiev, P. L. (Tchebycheff, Tschebysheff, etc.) 493 CebyHev system 218, 492, 518-9, qus. 6-10 for Ch. 4. Chapman-Kolrnogorov equations 496, 514
566
INDEX
Characteristic function 10-1, 17-8, 37-9, 56, 62-3,184-8,268-272,302,329,485, 51 2, distinguished from eigenfunction, 485, Chevalley, C. 488 Chihara, T. S. 494 Christoffel numbers 49 1 Christoffel-Darboux identity 31, 94, 98, 152, 173 Churchill, R. V. 482 Coddington, E. A. 478, 513-4, 521-2, 524 Cole, R. H. 523 Collatz, L. 478,481 Comparison theorems 523, 525, qu. 3, 7 for Ch. I , qu. 6 for Ch. 10, qu. 15 for Ch. 10, qus. 2-3 for Ch. 11. Conjugate points 300- I , 308-3 14, 328-9, 332-6 Completeness in hilbert space 471-5, 535 Conte, S. D. 522 Continued fractions 1 I 1, 492, 502 Courant, R. 478, 513, 520 Coxeter, H. S. M. 489 Crum, M. M. 529, 533 Cudov. L. A. 532 De Branges, L. 487 Definiteness 205, 253, 357, 374 Delsarte, J. 533 Determinants, of Jacobi form, 142-4, 501 Devinatz, A. 486, 501, 504, 513 Dickinson, D. J. 493, 494 Dickinson, D. R. 492 Difference equation (see also recurrence relation) 1, 2, 3 Differential equation, fourth-order 254-5, 323-8, 525-6, qus. 5 , 6, 8-19 for Ch. 10 first-order scalar 7, 55-7, 301-2 first-order vector-matrix, see under “system”, second-order, see under “Sturm-Liouville” Diffusion 495, 5 15, 5 18 Dikii, L. A. 482, 517 Disconjugacy 300-2, 525 Discrete approximation 3, 4, 8, 76-81, 359, 48 1, 530 Discrete-continuous problems 2, 8, 12, 76-81,452-6,481 in Sturm-Liouville theory, 226-8
Dolph, C. L. 478, 507, 521 Donohugh, W. F. Jr. 534 Duffin, R. J. 493 Eigenfunction expansion for first-order system of differential equations 261-2, 273-284 for integral equations 358-9, 41 1-5 for matrix recurrence relation 158 for scalar recurrence relations 34, 67-71, 105, 169, 175 for Sturm-Liouville equations 4, 222-6, 232-8, 243-7, 475, 481, 520 Eigenvalues, of varying matrices 303, 310, 317, 324-8, 335,337,457-470, 535 Eigenvalues, infinite 27-8, 24-6, 64 see also asymptotic behaviour Everitt, W. N. 502, 523, 535 Explicit expansion theorem I 17-8, 188-190, 240-3, qu. 12 for Ch. 6. Factorization 12, 192-3, 382, 51 1-3 Faddeev, L. D. 532 Favard, J. 493 Feller, W. 24, 495-6, 514, 527 Fort, T. 377,478,481,490-1, 501, 530 Fortet, R. 497 Frank, P. 518 Freud, G. 512 Frey, T. (Frei) 512 Friedman, A. 532 Fundamental solution 14, 29, 58-9, 94, 96, 253, 257, 264, 298, 362 see also “transfer-function” Gantmaher, F. R. (Gantmacher) 478, 490-2, 503, 518, 526 Gel’fand, I. M. 492, 524, 532-3 Geronimus, Ya. L. 479, 504, 512 Gilbert, R. C. 484, 521 Glazman, I. M. (Glasmann) 478, 484-5, 522-3,526,534 Gohberg, I. C. 513 Gol’dberg, A. A. 487 Good, I. J. 482 Gould, S. H. 492 Green’s function 10-1 I , 18-19, 145, 148-9, 229-232, 263, 266, 302, 358-9, 436, 50 1
INDEX
Green’s matrix 523 Green’s theorem 31, 98, 173 Greenstein, D. S. 500-1 Grenander, U. 479,504, 510 Gronwall, T. H. 455 Guillemin, E. A. 502 Halanai, A. 526 Hamburger, H. L. 486, 493, 501 Hannan, E. J. 479, 510-1 Harris, V. C. 516 Hartman, P. 490 Hausdorff, F. 486 Hellinger, E. 494-5 Helly-Bray theorem 61, 159, 299, 425-434, 534 Helson, H. 51 1 Herglotz, G. 508 Hestenes, M. R. 523 Hilbert, D. 478, 503, 513, 520 Hildebrandt, T. H. 527 Hille, E. 482 Hirschman, I. I. Jr. 519, 534 Hochstadt, H. 518 Howard, H. C. 526 Hunt, R. W. 526 Ince, E. L. 479, 503, 518 Ingram, W. H. 528 Initial-value problem 3 for integral equations 341-7 Integrable square 57, 250-1, 293-8, 391-3 see also “summable square” Integral equations 2, 521 with Stieltjes integrals 2 of Volterra-Stieltjes type 24, 204, 255, 339-415 of Fredholm-Stieltjes type 359, 515 Interface conditions 483, 516, 527-8 Interpolation in complex plane 71-4 on real axis 21 7-222 Invariance property 4-6, 13, 16, 83, 87, 93 Inverse problems 11-12, 39-46, 74, 107114, 178-181, 492, 531-3 Jackson, D. 489 Jeffreys, H. 8z B.S. 502 Jost, R. 532
567
Kac, I. S. 515, 520 Kamke, E. 479, 520 Karlin, S. 479, 490, 493, 503-4, 519 Kay, I. 532 Kazmin, Yu. A. 535 Kellogg, 0. D. 519 Kemp, R. R. D. 377, 530 Kemperman, J. H. B. 497 Klein, F. 503 Kodaira, K. 485, 521 Korop, V. F. 533 Krasnosel’skii, M. A. 479, 486, 500-1 Krein, M. G. 18, 199, 478-9, 486-7,490-3, 500-1, 503, 507, 513-5, 518-9, 522, 524, 526-7, 529-532 Kurzweil, J. 529 Kuz’mina, A. L. 512 Lagrange identity 31, 94, 98, 173, 348, 373 Lammel, E. 488 Langer, R. E. 482, 521 Leighton, W. 526 Lesky, P. 493 Levinson, N. 377, 478, 513-4, 521-2, 524, 530-2 Levitan, B. M. 479, 481, 491, 516, 521, 531-3 Lidskii, V. B. 524-5 Limit-circle, limit-point 15, 57, 63, 70, 129-130, 132, 134-5, 292-3, 299, 487, 495-7,499,501-2,521 Lions, J. L. 533 Liouville, J. 202, 342, 513, 518-9 Lipka, S. 492 Livlic, M. S. 485-6 Lowdenslager, D. 51 1 Lyapunov, M. A. 529 Lyubarskii, V. A. 524 Lyusternik, L. A. 535 MacNamee, J. 493 Mac Nerney, J. S. 502, 527-8 Mairhuber, J. C. 520 Mal’cev, A. I. 489 Mammana, G. 529 Mandl, P. 5 I5 MarEenko, V. A. 532 Marden, M.534 Markov, A. A. 493 Markov process 19, 496 Markov system 492
568
INDEX
Masani, P. 51 1 Matrix, contractive, 477 doubly-stochastic, 496 eigenvalues of, 457-470, 502, 535 harmonic, 528 hermitian, 476 imaginary part of, 477 J-contractive, 477, 488 J-unitary, 86, 253, 477, 488 Jacobi, 20, 142, 503 non-singular, 477 orthogonal, 477 positive-definite, 477 skew-hermitian 476 symmetric 4, 476 symplectic, 22, 86, 334, 477, 483, 488 McCarthy P. J. 491 McGregor, J. 494, 503-4, 519 McKean, H. P. Jr. 514, 516 McLeod, J. B. 517 McShane, E. J. 534 Mechanical quadrature 115, 121-2, 177, 49 1 Mikolis, M. 530 Mollerup, H. 519 Moment problem, 485-6, 500-1 for rational functions, 46-54, 71-74 Hamburger, 136-7, 184,486, 500 Hausdorff, 486, 501, 503 matrix, 502-3, 51 1 multi-dimensional 504 Stieltjes 486, 501 trigonometric, 184, 486, 508-9 Moore, R. A. 523 Morgan, G. W. 482 Moroney, R. M. 533 Morse, M. 479, 523-4 Moses, H. E. 532 Nagel, H. 503 Nagy, B. Sz. 479,502,514-5 Naimark, M. A. (Neumark) 479, 484-5, 513, 522, 531 Negative-imaginary function 436-440, 502 Nehari, 2,523,526 Nesting circles 125-7, 247-25 I , 284-9 299, 495 Networks, 2, 3, 5 , 10, 18, 19, 150, 502 Newton, R. G. 532
Nikolenko, L. D. 526 Non-oscillation, 300-2, 490, 495 Orthogonality, 441-446 dual, 33, 106, 175,483,496 Orthogonalization, continuous, 492, of polynomials, 108, 179, 492 of rational functions, 488 Oscillation theorem, multi-parameter, 164 on unit circle, 30, 173 Sturm-Liouville, 212 Oscillations, for differential equations, 300-338, 523526 for recurrence relations, 29, 100-4, 152-7 Parzen, E. 51 1 Peek, R. L. Jr. 482 Pick-Nevanlinna problem 73, 487 Pignani, T. J. 528 Plancherel, M. 481 Poisson, S. D. 513 Polar coordinate methods 101-2, 155-7, 210-7, 303, 305, 329, qus. 7, 8, 16 for Ch. 8, qu. 5 for Ch. 12 Pollak, H. 0. 493 Polynomials, classical, Hermite, 493, qu. 8 for Ch. 5 , Jacobi, 493 Laguerre, 493, qu. 10 for Ch. 5, Legendre, 118, 135, 493, qu. 3 for Ch. 5. Polynomials, orthogonal on real axis, 16,9092, 97-141, 213, 487, 489-491, 493-4 asymptotic behaviour, 196-8, 493, 51 2 as determinants, 142-9, 501 in several variables, 160-9, 503-4 quasi-orthogonal, 494 with matrix coefficients, 150-160, 254, 262, 497, 502 Polynomials orthogonal on the unit circle, 92-3, 96, 170-201 Porter, M. B. 490 Positive-real functions 436, 534 Potapov, V. P. 479, 488, 528 Prediction 510-1 Price, G. B. 528 Probability 19, 482,497, 503, 505 Priifer, H. 222, 358, 520, 523
569
INDEX
Quasi-differential equation 203, 255, 522 Queues 508
for polynomials on real axis 101-4,145-8 for polynomials on unit circle 186 for rational functions 30 Random variable 509,510 Seshu, S. and L.534 Random walk 19,505-9 Shear 91,489 Ray, D.B. 516 Sherman, J. 494 Recurrence relations 2-4, 7-1 1, 15-21, Shohat, J. 479,494,500, 504,534 Siege], C. L.489 25-203,487, 489-491,493-5,501-2 multivariate 160,qu. 8 for Ch. 1 Sims, A. R. 495,521 oscillatoryproperties29-30,100-104,4901 Spectral function 10, 35-7, 46-51, 56, with matrix coefficients 83-96,150-160, 60-2, 120-5, 182-4, 238-240, 262,
502-3
within
differential
equations,
202-3,
254-5,516
Rehtman, P. G. 479,500 Reid, W.T. 516,522-7 Resolvent equation 265 Resolvent kernel 263-8 Riccati equation 209, 302 matrix, 304, 325, 329,523 Richardson, J. M. 492 Riesz, F.479,484 Right-conjugate 300-1,308, 324, 329 Rooney, P. G. 508 Roos, B.W.522 Rosen, J. B. 534 Rosenblatt, M.51 1 Rosenfeld, N. S . 518 Rota, G.-C. 475,478, 513, 522, 533, 535 Rozanov. Yu. A. 511 S-function 371-383 S-matrix 532 Sandor, St. 526 Sangren, W. C. 483,520,522, 528 Sargsyan,,I. S . 516, 521 Scarpellini, B. 519 Scattering 377, 532 Scharf, H.528 Schmidt, T.W.493 Schoenberg, I. J. 520,525 Schur, I. 503 Schwarz, B. 492 Schwerdtfeger, H.488 Self-adjoint 4, 498-500 Separation theorems 17, 300,302 for first-order system 332-7 for matrix polynomials 156-7 for matrix Sturm-Liouville systems
308-312,318, 523-4
500-1 298,382-3,483-5,487-8,494-6, 520-1 limiting 129-130, 139, 496 multi-dimensional 169 orthogonal 485, 496, 501 Spectral kernel 484 Spectrum 27, 70,371,490
Spencer, V. E. 501 Spitzer, F. 506 Stability 447-456 for differential equations 206, 449-452 for integral equations 356, 368, 384,
452-6
for recurrence relations 131-2,447-9 Stallard, F. W.527-8 Stieltjes integral 2, 35, 204, 416-435,445,
455, 534
modified, 360, 363-4,528 Stieltjes, T. J. 486,493 Stochastic process 509 Stone, M. H.479,484,489,494, 500, 531,
534
Stratton, J. A. 481 Straus, A. V. 485, 488 String, vibrating 2, 5, 16,21,24, 119, 148,
202, 492-4,497, 501, 514, 516
Sturm, Ch. 202,218,490,513, 518 Sturm-Liouville theory 17, 21-4,30, 97,
147, 150, 160, 202-254, 261, 263, 338, 340, 355, 381-3,398-415,440, 473, 475, 482-3,489, 513-521 discrete, 4, 97-149 eigenfunction expansion 4,222-6,358-9, 411-5 inverse, 485,492, 530-3 matrix, 254, 300,303-323 on a half-axis, 243-251,487, 512, 521 Summable sauare 60. 130-2.484,495 see also “integrable square”
570
INDEX
Symplectic transvection 91, 489 System of differential equations 13-14, 252-300, 332-8, 521-2 bounds for eigenvalues 256, 338 conjugate points, 300, 332-6 eigenfunction expansion, 273-284 generalised to integral equations, 359-365 nesting circles, 284-9 over infinite intervals, 289-299 Szisz, 0.488 Szegb, G. 479,480,489-491, 493, 501, 504, 510-2, 530 Tamarkin, J. D. 479, 482, 500, 504, 534 Tarnopol’skii, J. G. 493, 500 Tchebychev, Tschebysheff, etc. see “Cebyiev” Tensor product 503 Teptin, A. L. 501 Time series 509 Titchmarsh, E. C. 480,484, 497, 501. 513, 516, 521-2 Toeplitz form 184, 199, 509 Trace, of matrix 476 Trace-formulae for eigenvalues 380, 399, 51 7 Tricomi, F. G. 489, 493 Truxal, J. G. 481
ViHik, M. I. 535 von Mises, R. 518 Wall, H. S. 480, 495, 500, 528 Walsh, J. L. 533 Wannier, G. H. 493 Wave propagation 3,5-6,481 Wax, N. 518 Wendel, J. G. 506 Wendroff, B. 491 Weyl, H. 487,495 Whyburn, W. M. 490, 517-8, 523, 528 Widder, D. V. 480, 500, 519, 534 Wiener, N. 506, 51 I Wiener-Hopf equation 504, 507 Williamson, R. E. 520 Wintner, A. 490, 525 WKB-method, 518 Wronskian, 4, 6, 28, 100, 490, 529, 530 for integral equation, 348-350 Wylie, C. R. Jr. 520 YakuboviE, V. A. 489, 524 Zarhina, R. B. 504 Zimmerberg, H. J. 482