The Theory of Splines and Their Applications (Mathematics in Science and Engineering 38)

The Theory of Splines and Their Applications

M A T H E MAT I C S I N SCIENCE AND ENGINEERING ~~

A SERIES OF M O N O G R A P H S A N D T E X T B O O K S

Edited by Richard Bellman University of Southern California 1.

2. 3. 4.

5. 6. 7. 8. 9.

i0. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22.

TRACY Y. THOMAS. Concepts from Tensor Analysis and Differential Geometry. Second Edition. 1965 TRACY Y.THOMAS. Plastic Flow and Fracture in Solids. 1961 RUTHERFORD ARIS. The Optimal Design of Chemical Reactors: A Study in Dynamic Programming. 1961 JOSEPH LASALLEand SOLOMON LEFSCHETZ. Stability by Liapunov’s Direct Method with Applications. 1961 GEORGE LEITMANN (ed.) . Optimization Techniques : With Applications to Aerospace Systems. 1962 RICHARDBELLMANand KENNETHL. COOKE.Differential-Difference Equations. 1963 Mathematical Theories of Traffic Flow. 1963 FRANKA. HAIGHT. F. V. ATKINSON. Discrete and Continuous Boundary Problems. 1964 Non-Linear Wave Propagation: With AppliA. JEFFREY and T. TANIUTI. cations to Physics and Magnetohydrodynamics. 1964 JULIUS T. Tou. Optimum Design of Digital Control Systems. 1963 HARLEY FLANDERS. Differential Forms: With Applications to the Physical Sciences. 1963 SANFORD M. ROBERTS. Dynamic Programming in Chemical Engineering and Process Control. 1964 SOLOMON LEPSCHETZ. Stability of Nonlinear Control Systems. 1965 DIMITRISN. CHORAFAS. Systems and Simulation. 1965 A. A. PERVOZVANSKII. Random Processes in Nonlinear Control Systems. 1965 MARSHALL C. PEASE,111. Methods of Matrix Algebra. 1965 V. E. BENES.Mathematical Theory of Connecting Networks and Telephone Traffic. 1965 WILLIAM F. AMES.Nonlinear Partial Differential Equations in Engineering. 1965 J. A C Z ~ LLectures . on Functional Equations and Their Applications. 1966 R. E. MURPHY.Adaptive Processes in Economic Systems. 1965 S. E. DREYFUS.Dynamic Programming and the Calculus of Variations. 1965 A. A. FEL’DBAUM. Optimal Control Systems. 1965

MATHEMATICS 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

IN S C I E N C E A N D ENGINEERING

A. HALANAY. Differential Equations: Stability, Oscillations, Time Lags.

1966 M. NAMIKOGIUZTORELI. Time-Lag Control Systems. 1966 DAVIDSWORDER. Optimal Adaptive Control Systems. 1966 MILTONASH. Optimal Shutdown Control of Nuclear Reactors. 1966 DIMITRIS N. CHORAFAS. Control System Functions and Programming Approaches. (In Two Volumes.) 1966 N. P. ERUGIN.Linear Systems of Ordinary Differential Equations. 1966 SOLOMON MARCUS.Algebraic Linguistics ; Analytical Models. 1967 A. M. LIAPUNOV. Stability of Motion. 1966 GEORGELEITMANN (ed.). Topics in Optimization. 1967 MASANAO AOKI. Optimization of Stochastic Systems. 1967 HAROLD J. KUSHNER.Stochastic Stability and Control. 1967 MINORUURABE.Nonlinear Autonomous Oscillations. 1967 F. CALOGERO. Variable Phase Approach to Potential Scattering. 1967 A. K A U F M A N Graphs, N. Dynamic Programming, and Finite Games. 1967 A. K A U F M A Nand N R. CRUON. Dynamic Programming: Sequential Scientific Management. 1967 J. H. AHLBERG, E. N. NILSON,and J. L. WALSH.The Theory of Splines and Their Applications. 1967

In prepavation Y . SAWARAGI, Y . S U N A ~ A Rand A , T. NAKAMIZO. Statistical Decision Theory in Adaptive Control Systems A. KAUFMANN and R. FAURE.Introduction to Operations Research RICHARD BELLMAN. Introduction to the Mathematical Theory of Control Processes ( I n Three Volumes.) E. STANLEY LEE. Quasilinearization and Invariant Bedding WILLARDMILLER,JR. Lie Theory and Special Functions F. SHAMPINE, and PAULE. WALTMAN. Nonlinear PAULB. BAILEY,LAWRENCE Two Point Boundary Value Problems

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The Theory of Splines and Their Applications J . H . AHLBERG UNITED AIRCRAFT RESEARCH LABORATORIES EASTHARTFORD, CONNECTICUT

E . N . NILSON PRATT & WHITNEY AIRCRAFT COMPANY EASTHARTFORD, CONNECTICUT

J . L. WALSH UNIVERSITY OF MARYLAND COLLEGE PARK,MARYLAND

1967

ACADEMIC PRESS

New York and London

COPYRIGHT 0 1967,

BY

ACADEMIC PRESSINC.

ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 1 1 1 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 66-30115

PRINTED I N THE UNITED STATES OF AMERICA

PREFACE

Spline functions constitute a relatively new subject in analysis. During the past decade both the theory of splines and experience with their use in numerical analysis have undergone a considerable degree of development. Discoveries of new and significant results are of frequent occurrence. It is useful at this juncture, nevertheless, to make some serious effort to organize and present material already developed up to this time. Much of this has become standardized. On the other hand, there are several areas where the theory is not yet complete. This book contains much of the material published since 1956 together with a considerable amount of the authors’ own research not previously presented; it also reflects a considerable amount of practical experience with splines on the part of the authors. I n the interests of holding the present volume to a reasonable size, certain areas related to splines have been omitted. T h u s the work of Schoenberg and his associates o n the use of splines in the smoothing of equidistant data has not been included, nor is there any treatment of the theory of splines of complex argument. We hope, nevertheless, that the material presented will provide the reader with the necessary background for both theoretical and applied work in what promises to be a very active and extensive area. I n Chapter I there is a brief description of what is meant by a spline; this is followed by a survey of the development of spline theory since 1946 when Schoenberg first introduced the concept of a mathematical spline. We develop in Chapters I1 and IV, respectively, the theory of cubic splines and polynomial splines of higher degree from an algebraic point of view; the methods employed depend heavily on the equations used to define the spline. I n particular, these chapters contain much of the material basic for applications. I n Chapters I11 and V we reconsider cubic and polynomial splines of higher degree from a different point of view which reveals more clearly their deeper structure. Although the resulting theorems are not so sharp as their counterparts in Chapters 11 and IV, they are more easily carried over to new settings. This is done vii

...

Vlll

PREFACE

in Chapters VI, VII, and VIII, in which we consider in turn generalized splines, doubly cubic splines, and two-dimensional generalized splines. We wish to express our deep gratitude to all those who have contributed to making this book a reality. Specifically, we wish to thank the United Aircraft Research Laboratories, the Pratt & Whitney Division of the United Aircraft Corporation, Harvard University, and the University of Maryland, whose support has made possible much of our research in spline theory.

May, 1967

J. H. AHLBERG E. N. NILSON J. L. WALSH

CONTENTS

PREFACE.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Introduction

Chapter I

1.1. What Is a Spline ? . . . . . . . . . . . . . . 1.2. Recent Developments in the Theory of Splines

Chapter I1 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

Chapter I11

...... . . . . . . .

The Cubic Spline Introduction . . . . . . . . . . . . . . . . . . . . . .

Existence. Uniqueness. and Best Approximation . Convergence . . . . . . . . . . . . . . . . Equal Intervals . . . . . . . . . . . . . . . Approximate Differentiation and Integration . . . Curve Fitting . . . . . . . . . . . . . . . Approximate Solution of Differential Equations . Approximate Solution of Integral Equations . . Additional Existence and Convergence Theorems

. . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Intrinsic Properties of Cubic Splines The Minimum Norm Property . . . . . . . . . . . . . . The Best Approximation Property . . . . . . . . . . . . . The Fundamental Identity . . . . . . . . . . . . . . . . The First Integral Relation . . . . . . . . . . . . . . . . Uniqueness . . . . . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . General Equations . . . . . . . . . . . . . . . . . . . Convergence of Lower-Order Derivatives . . . . . . . . . The Second Integral Relation . . . . . . . . . . . . . . . .

3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. Raising the Order of Convergence . . . . 3.1 1. Convergence of Higher-Order Derivatives 3.12. Limits on the Order of Convergence . . . 3.13. Hilbert Space Interpretation . . . . . . 3.14. Convergence in Norm . . . . . . . . . 3.15. Canonical Mesh Bases and Their Properties 3.1 6. Remainder Formulas . . . . . . . . . . 3.17. Transformations Defined by a Mesh . . . 3.18. A Connection with Space Technology . . ix

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . . . . . . . . . . .

1 2

9 16 19 34 42 50 52 57 61

75 77 78 79 82 84 84 87 89 91 93 95 97 98 101 103 105 107

CONTENTS

X

Chapter I V 4.1. 4.2. 4.3. 4.4. 4.5. 4.6.

The Polynomial Spline Definition and Working Equations . . . . . . . . . . . . Equal Intervals . . . . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . Convergence . . . . . . . . . . . . . . . . . . . . . . Quintic Splines of Deficiency 2, 3 . . . . . . . . . . . . . Convergence of Periodic Splines on Uniform Meshes . . . . .

109 124 132 135 143 148

Intrinsic Properties of Polynomial Splines of Odd Degree

Chapter V 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.1 1. 5.12. 5.13. 5.14. 5.15. 5.16. 5.17. 5.18.

Chapter VI 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10. 6.1 1. 6.12. 6.13. 6.14. 6.15. 6.16.

Introduction . . . . . . . . . . . . . . . . . . . . . . The Fundamental Identity . . . . . . . . . . . . . . . The First Integral Relation . . . . . . . . . . . . . . . The Minimum Norm Property . . . . . . . . . . . . . The Best Approximation Property . . . . . . . . . . . . Uniqueness . . . . . . . . . . . . . . . . . . . . . . Defining Equations . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . Convergence of Lower-Order Derivatives . . . . . . . . The Second Integral Relation . . . . . . . . . . . . . . Raising the Order of Convergence . . . . . . . . . . . . Convergence of Higher-Order Derivatives . . . . . . . . Limits on the Order of Convergence . . . . . . . . . . . Hilbert Space Interpretation . . . . . . . . . . . . . . Convergence in Norm . . . . . . . . . . . . . . . . . Canonical Mesh Bases and Their Properties . . . . . . . Kernels and Integral Representations . . . . . . . . . . Representation and Approximation of Linear Functionals . .

153 154 155 156 157 159 . 160 165 . 166 . 168 . 170 . 172 . 174 . 174 . 176 . 179 . 182 . 185

. . . .

Generalized Splines Introduction . . . . . . . . . . . . . . . . . . . . . . The Fundamental Identity . . . . . . . . . . . . . . . . The First Integral Relation . . . . . . . . . . . . . . . . The Minimum Norm Property . . . . . . . . . . . . . . Uniqueness . . . . . . . . . . . . . . . . . . . . . . Defining Equations . . . . . . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . . . . . . . . . Best Approximation . . . . . . . . . . . . . . . . . . Convergence of Lower-Order Derivatives . . . . . . . . . The Second Integral Relation . . . . . . . . . . . . . . . Raising the Order of Convergence . . . . . . . . . . . . . Convergence of Higher-Order Derivatives . . . . . . . . . Limits on the Order of Convergence . . . . . . . . . . . . Hilbert Space Interpretation . . . . . . . . . . . . . . . Convergence in Norm . . . . . . . . . . . . . . . . . . Canonical Mesh Bases . . . . . . . . . . . . . . . . .

191 192 193 195 196 197 199 200 201 204 206 208 211 213 216 219

xi

CONTENTS

6.17. Kernels and Integral Representations . . . . . . . . . . . 6.18. Representation and Approximation of Linear Functionals . . .

Chapter VII 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.1 1. 7.12. 7.13. 7.14. 7.15. 7.16,

220 221

The Doubly Cubic Spline Introduction . . . . . . . . . . . . . . . . . . Partial Splines . . . . . . . . . . . . . . . . . Relation of Partial Splines to Doubly Cubic Splines The Fundamental Identity . . . . . . . . . . . The First Integral Relation . . . . . . . . . . . The Minimum Norm Property . . . . . . . . . Uniqueness and Existence . . . . . . . . . . . Best Approximation . . . . . . . . . . . . . Cardinal Splines . . . . . . . . . . . . . . . . . Convergence Properties . . . . . . . . . . . . The Second Integral Relation . . . . . . . . . . T h e Direct Product of Hilbert Spaces . . . . . . The Method of Cardinal Splines . . . . . . . . . Irregular Regions . . . . . . . . . . . . . . . . Surface Representation . . . . . . . . . . . . The Surfaces of Coons . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

.

.

. .

235 237 238 240 242 242 243 244 245 247 248 249 251 254 258 262

Chapter VIII Generalized Splines in Two Dimensions 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.

. . . . . . . . 265 . . . . . . . . 266 . . . . . . . . . 267 . . . . . . . . 269 . . . . . . . . . 270 . . . . . . . . 271 . . . . . . . . 272 . . . . . . . . 274 . . . . . . . . . 275

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

281

Bibliography

INDEX.

Introduction . . . . . . . . . . . . . . Basic Definition . . . . . . . . . . . . The Fundamental Identity . . . . . . . Types of Splines . . . . . . . . . . . . The First Integral Relation . . . . . . . Uniqueness . . . . . . . . . . . . . . Existence . . . . . . . . . . . . . . . . Convergence . . . . . . . . . . . . . . Hilbert Space Theory . . . . . . . . .

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CHAPTER I

Introduction

1.1. What Is a Spline?

It seems appropriate to begin a book on spline theory by defining a spline in its simplest and most widely used form, and also to indicate the motivation leading to this definition. For many years, long, thin strips of wood or some other material have been used much like French curves by draftsmen to fair in a smooth curve between specified points. These strips or splines are anchored in place by attaching lead weights called “ducks” at points along the spline. By varying the points where the ducks are attached to the spline itself and the position of both the spline and the duck relative to the drafting surface, the spline can be made to pass through the specified points provided a sufficient number of ducks are used. If we regard the draftsman’s spline as a thin beam, then the BernoulliEuler law M ( x ) = EI[l/R(x)]

is satisfied. Here M ( x ) is the bending moment, E is Young’s modulus,

I is the geometric moment of inertia, and R ( x ) is the radius of curvature of the elastica, i.e., the curve assumed by the deformed axis of the beam. For small deflections, R(x) is &placed by l/y”(x),where y ( x ) denotes the elastica. T hu s we have y”(x) = (l/EI)M(x).

Since the ducks act effectively as simple supports, the variation of M ( x ) between duck positions is linear. T h e mathematical spline is the result of replacing the draftsman’s spline by its elastica and then approximating the latter by a piecewise cubic (normally a different cubic between each pair of adjacent ducks) with certain discontinuities of derivatives permitted at the junction points (the ducks) where two cubics join. 1

2

I.

INTRODUCTION

I n its simple form, the mathematical spline is continuous and has both a continuous first derivative and a continuous second derivative. Normally, however, there is a jump discontinuity in its third derivative at the junction points. This corresponds to the draftsman’s spline having continuous curvature with jumps occurring in the rate of change of curvature at the ducks. For many important applications, this mathematical model of the draftsman’s spline is highly realistic. I n practice, the draftsman does not place the ducks at the specified points through which his splin’e must pass. Moreover, there is not usually a one-to-one correspondence between the specified points and the ducks. On the other hand, when the mathematical analog is used, it is common practice to interpolate to the specified points at the junction points and to keep the number of specified points and junction points (including the endpoints) the same. I n the next section, we outline the recent history of the mathematical spline approximation. From this history, some of the properties of the mathematical spline become evident. Also, a considerable extension of the concept of a spline from that approximating the draftsman’s tool is apparent.

1.2. Recent Developments in the Theory of Splines T h e spline approximation in its present form first appeared in a paper by Schoenberg [ 19461.* As indicated in Section 1 . 1 , there is a very close relationship between spline theory and beam theory. Sokolnikoff [1956, pp. 1-41 provides a brief but very readable account of the development of beam theory. From the latter, one might anticipate some of the recent developments in the theory of splines, particularly the minimum curvature property. As suggested in Schoenberg’s paper [ 19461, approximations employed in actuarial work also frequently involve concepts that relate them closely to the spline. After 1946, Schoenberg, together with some of his students, continued these investigations of splines and monosplines. I n particular, Schoenberg and Whitney [1949; 19531 first obtained criteria for the existence of certain splines of interpolation. For the case of splines of even order with interpolation at the junction points, a simpler approach to the question of existence due to Ahlberg, Nilson, and Walsh [1964; 19651 is now possible; it makes use of a basic integral relation obtained for cubic

*

Data in square brackets refer to items in the Bibliography.

1.2.

RECENT DEVELOPMENTS I N THE THEORY OF SPLINES

3

splines of interpolation to a function f ( x )on a mesh A by Holladay [1957] which asserts

I” a

If”(x)

l2

dx =

a

1 S i ( f ;x) la dx

+ J’ If”(x) a

-

S i ( f ;x) l a dx.

Here S,( f ; x) denotes the spline of interpolation to f f x ) on A . I n this book, we refer to this integral relation as the first integral relation. T h e establishment of the first integral relation for certain cubic splines of interpolation was Holladay’s proof of the following theorem.

Theorem (Holladay). Let A : a = x,, < x1 < < xN = b and a set of real numbers {yi} (i = 0, I , ..., N ) be given. Then of all functions f ( x ) having a continuous second derivative on [a, b] and such that f ( x i ) = yi (i = 0, I , ..., N ) , the spline function S,( f ; x) with junction points at the xi and with Si(f ; a) = Si(f;6 ) = 0 minimizes the integral -.a

(1.2.1)

Much of the present-day theory of splines began with this theorem and its proof. Since the integral (1.2.1) is often a good approximation to the integral of the square of the curvature for a curve y = f ( x ) , the content of Holladay’s theorem is often called the minimum curvature property. Its close relation to the minimization of potential energy of a deflected beam is apparent. * I n this book, we consider a number of generalizations of the simple cubic spline. I n these generalizations, there are analogs of Holladay’s theorem; but, since there is no relation to curvature in these new settings, we use the name minimum norm property instead. This is meaningful, since in each case there is an associated Hilbert space, for now denoted by 2,in which (1.2.1) or its counterpart is the square of the norm of f ( x ) . It was not until 1964 that the Hilbert space aspect of spline theory evolved. At that time, the authors (Ahlberg, Nilson, and Walsh [abs. 1964; 19641) introduced some orthonormal bases for the space 2 which consisted entirely of splines or, somewhat more precisely, of equivalence classes of splines. I n terms of any orthonormal basis for 2, a functionffx) in % has, of course, for any positive integer N , a best approximation by linear combinations of the first N basis elements.

* T h e potential energy of a statically deflected beam is equal to the work done on the beam to produce the deflections; this in turn is proportional to the integral of the square of the curvature of the elastica of the beam (cf. Sokolnikoff [1956, p. 21).

4

I.

INTRODUCTION

If II * llx denotes the norm of 2, Ui(x) (i = 1, 2, ...) denotes the basis elements, and

U(4

N

=

2a,Ui(x),

(1.2.2)

i=l

then l l f - U is minimized when ai is the coefficient of U,(x) in the expansion of f ( x ) in terms of the complete basis. It is desirable to have another characterization of this best approximation, particularly if the alternative characterization facilitates its determination. Such a characterization is available. I n 1962, the authors (Walsh, Ahlberg, and Nilson [1962]) obtained the result: given a mesh A : a = x, < x1 < < x N = b, then of all simple periodic cubic splines on A the spline that interpolates to a periodic function f ( x ) at the mesh points furnishes the best approximation in the preceding sense. Since then, a number of extensions of this result have been obtained: Ahlberg, Nilson, and Walsh [abs. 1963; abs. 1964a,b; 1964; 19651, deBoor [1963], Schoenberg [1964c], Greville [1964a], and deBoor and Lynch [abs. 1964; 19661. I n Chapters 111, V, and VI of this book, we develop the HiIbert space theory of splines for cubic splines, polynomial splines of odd degree, and generalized splines, respectively. We define 2 as a function space of classes of functions; we then show that 2 is a Hilbert space with respect to an appropriate choice of norm. T h e symbol 2, however, is replaced by other notation. T h e convergence of the spline approximations Sy)(f ;x ) to the as the mesh norm jl A /I = maxi approximated functionsf - xj I approaches zero has also come under close scrutiny. T h e first results were obtained by the authors (Walsh, Ahlberg, and Nilson [1962]) for cubic splines and utilized the first integral relation. Under the assumption that f ( x ) is in C2[a,b], it was shown for splines of interpolation to f ( x ) at the mesh points that S y ) ( f ;x ) converges uniformly to f ' e ) ( ~ for ) 01 = 0, 1. A more detailed analysis was made by Ahlberg and Nilson [abs. 1961; 1962; 19631. I n particular, it was shown that, if f ( x ) is in C2[a,b], then Si(f;x) converges uniformly to f " ( x )provided the mesh spacing approaches uniformity as I/ A 11 approaches zero. This mesh restriction was later removed by Sharma and Meir [abs. 1964; 19661. For f ( x ) in C4[a,b], Birkhoff and deBoor [abs. 1964; 19641 have shown that (1.2.3) .( = 0, 1, ..., 4) i p ( x ) - sy(f;X) 1 < K IA ~114-= provided the ratio Rd = maxill A / I / xi - xiPl I is bounded. On the other hand, for weaker restrictions on f ( x ) such as f ( x ) is in C[a, b] or

1.2.

RECENT DEVELOPMENTS I N THE THEORY OF SPLINES

5

f(x) is in Cl [ a,b], appropriate convergence properties have been obtained by Ahlberg, Nilson, and Walsh [abs. 19661. I n addition, the convergence of polynomial splines of odd degree has been investigated by Ahlberg, 'Nilson, and Walsh Cabs. 1963; 19651, Schoenberg [1964], and Ziegler [abs. 19651; the convergence of multidimensional splines by Ahlberg, Nilson, and Walsh [abs. 1964a; 1964; 1965al; and the convergence of generalized splines by the same authors [abs. 1964b; 1964; 1965a1. Many of these convergence results depend on the fine structure of the linear system of equations defining the spline. I n Chapters I1 and IV, we develop spline theory from this point of view. On the other hand, a number of convergence results can be established without appeal to the defining equations. I n particular, for polynomial splines of degree 2n - 1 this can be done with respect to the convergence of derivatives through order n - 1. Moreover, with the aid of the integral relation

which was established by Ahlberg, Nilson, and Walsh [1965a] under a variety of conditions, convergence of derivatives through order 2n - 2 can be established. This was shown for generalized splines, polynomial splines of odd degree being a special case of the latter. We refer to the integral relation (1.2.4) as the second integral relation; on a suitable function space, it is a manifestation of the Riesz theorem concerning the representation of linear functionals. I n Chapters 111, V, and VI, this approach is developed. T h e basic result is -

S y ) ( f ;x) 1

< K 11 d 1 2n-a-1

(a = 0,

1 ,..., 2n - 1)

(1.2.5)

provided RA is bounded. For cubic splines, this result is weaker than (1.2.3). Whether 2n - 01 - 1 can be replaced in general by 2n - 01 is an open question. T h e theory of splines has been extended in a number of directions. Of considerable importance is the extension to several dimensions. A start was made by Birkhoff and Garabedian [1960], but the first truly successful extension was made by deBoor [abs. 1961; 19621, who demonstrated (deBoor [1962]) both the existence and uniqueness of certain bicubic splines* of interpolation. Later Ahlberg, Nilson, and Walsh [abs. 1964a; 1965bI extended the first integral relation to splines in several dimensions. As a result, existence, uniqueness, the minimum

* We employ henceforth the terminology j f t , s) is Q doubte cubic rather thanf(t, s) is a bicubic to imply thatf(t, s) is a cubic in t for each s and a cubic in s for each t . A doubZy cubic s p h e is a double cubic in each subrectangle defined by a two-dimensional mesh.

6

I.

INTRODUCTION

norm property, and the best approximation property were obtained for a variety of multidimensional splines. Questions of convergence were reduced to similar questions in one dimension, for which answers were known. I n Chapters VII and VIII, we consider multidimensional splines. Another direction of generalization has been the replacement of the operator DZn associated with a polynomial spline of degree 2n - 1 (here D z dldx) by the operator L*L, where

and L* is the formal adjoint of L. I n each mesh interval, a spline S ( x ) now satisfies the equation L*LS = 0 rather than the equation D2nS = 0. Splines defined in this manner are called generalized splines. T h e first step in this direction was taken by Schoenberg [I 964~1,who considered L< trigonometric splines." T h e complete generalization followed: Greville [1964]; Ahlberg, Nilson, and Walsh [abs. 196413; 1964; 1965al; deBoor and Lynch [abs. 19641. A more abstract approach to spline theory has been made by Atteia [1965] and his colleagues at Grenoble. T h e operator L = D(D -- u) has been considered by Schweikert [1966], who termed the resulting splines "splines in tension." When u is properly chosen, these splines have some advantages over cubic splines, as well as some disadvantages; in particular, they tend to suppress the occurrence of inflexion points not indicated by the data but concentrate the curvature near the junction points. Generalized splines are the subject matter of Chapters VI and VIII. T h e approximation of a linear functional9 by a second linear functional 9 such that the remainder W = 9 - 2 annihilates polynomials of degree n - 1 has been given considerable attention by Sard [1963]. Under reasonable restrictions, (1.2.7)

the kernel X ( 9 ;t ) is called a Peano kernel. Sard [1963] has sought to determine 9 such that jlX(B?;t ) 2 dt

is minimized, and for a variety of functionals 9he has so determined 9. In 1964, Schoenberg succeeded in showing that, for 9 of the form

9 f = @of(Jco) + .J(Jcl>

+ ..-+

1.2.

RECENT DEVELOPMENTS I N THE THEORY OF SPLINES

7

and with mild restrictions on 2,the optimum 9 results when L?j = 9 S d ( f ;x), whereS,(f; x) is the simple(see be1ow)polynomial spline (of degree 2n - 1) interpolating tof(x) onfl: a = xo < x1 < < xN = 6 . This result has been generalized (Ahlberg, Nilson, and Walsh [abs. 19651; Ahlberg and Nilson [1966]) to &’ of the form

p < n - 2 and certain a,,j = 0 ab initio. For generalized splines, (1.2.7) becomes

with

9f=

b

X ( 9 ;t)Lf(t)dt,

a

and the kernel X ( & ;t ) depends on L (cf. also deBoor and Lynch [1966]). We consider these matters in Chapters 111, V, VI, and VIII. T h e generalization of Schoenberg’s results for approximating functionals required the introduction of splines of a somewhat different character. T h e following terminology facilitates a partial indication of the nature of these differences. A spline of order 2n is simple when there is at most a jump discontinuity in the (2n - 1)th derivative at a mesh point. I n most instances, the splines under consideration are simple splines. When jumps in derivatives of order greater than 2n - k - 1 are permitted at an interior mesh point xi , the spline is said to be of dejiciency k at xi. If the spline is of deficiency k at all interior mesh points, it is said to be of deficiency k. We impose, however, the restriction 0 k n. I n this terminology, a solution of L*Lf = 0 in [a, b] has deficiency zero, and a simple spline has deficiency one. T h e requirement that certain aij in (1.2.8) vanish ab initio often imposes even more complicated and irregular continuity requirements on the splines employed. Such splines are called heterogenous splines and are considered in detail in Chapters VI and VIII. They were introduced by Ahlberg, Nilson, and Walsh [abs. 19651 for studying the approximation of linear functionals. T h e work of Golomb and Weinberger on “optimal’’ approximation of linear functionals (cf. Golomb and Weinberger [1959]) is very closely related to spline theory. I n many instances the functions zi(x) entering

<
h>O

is equivalent to placing a simple support at x-, = (xo - Ax,)/( 1 - A) and requiring that the entire curve over x-, x x1 be the arc of a cubic. A common choice of A is i. We are generally concerned with end conditions, which, for convenience, we write in the form

<


b:

(2.1.1 9)

2mN = 2 y i ,

+ 2mN = 3 ANmN-, + 2mN = cN . mNPl

Here y i and y k are prescribed values of the spline second derivative at the ends of the interval. Moreover, these two sets are equivalent provided

It is readily verified that the right-hand members of (2.1.7) and (2.1.15) are three times the values of the second and first derivatives at xi, respectively, of the parabola through (xipl,y i P l ),( x i ,yj), and (xi+l ,y j t l ) having vertical axis. This fact has special significance relative to the problem of curve fitting with higher-order splines and is examined in detail in Chapter IV. A very efficient algorithm is available for solving the system of equations (2.1.8) or (2.1.16). Given the equations

+ + bZx2 + + + + + b,x,

alxl

~ 3 x 2

an-1xn-2

b3x3

bn-lXn-1

anxn-1

form ( K

=

1, 2, ..., a )

c1x2

= dl

~ 2 x 3=

4

, 1

~ 3 x 4= d3 >

...

Cn-lxn = 4 - 1

+

bnxn =

dn

9

( 2 .I .20)

2.1.

15

INTRODUCTION

Successive elimination of x1 , x2 ,..., xnP1 from 2nd, 3rd ,...,nth equations yields the equivalent equation system Xk = qkxk+l xn = U n

+ Uk

(k

=

1, ..., It

-

l),

(2.1.21)

7

whence x,, x , _ ~ ,..., x, are successively evaluated. For matrices with dominant main diagonil, with which we are primarily concerned, this procedure is stable in the sense that errors rapidly damp out (0 < ck/pk < 1). We note also that the quantities p , and qk in the application to the spline depend upon the mesh A but not upon the ordinates at the mesh locations. Thus, several spline constructions on the same mesh may be carried out with the computation of only one set of p,’s and q,’s. An extension of this procedure is used for the periodic spline. For the equations

an-1~n-2

+ + + + + + + + bixi

~1x2

aixn =

~2x1

b2x2

~ 2 x 3=

bn-lxn-1

cnx1

...

4, 4 9

cn-ixn = 4 - 1

anxn-1

bnxn =

dn

9

9

we effectively solve for x, ,..., x,-~ in terms of x, by means of the first n - 1 equations and then determine x, from the last equation. In addition to the quantities p , , qk , ukdetermined previously, we therefore calculate for k = 1, 2, ..., n the quantities Sk

=

(so =

-Wk-l/pk

1)

(2.1.22)

so that Eqs. (2.1.21) are replaced by the relations Xk = qkxk+l

+

SkXn

+

01,

+ Uk

(k

=

1 , ..., n - 1).

If we write the equation Xk = t k X ,

then we have tk

= qktk+l

vk

= qkvk+l

(k = 1 , 2,..., n

+ +

-

(2.1.23)

I),

I), ( % = 0). (&

sk

uk

=

We determine tkpl ,..., t , , vkPl ,..., v, and evaluate x, from the equation cn(t1xn

+ + 01)

an(tn-ixn

+

wn-1)

We then determine xnP1 ,..., x1 from (2.1.23).

+

bnxn =

4

*

11.

16

THE CUBIC SPLINE

We note that, when a k , b,, and c, are all constant, the quantities and q, can be obtained from a solution of a second-order difference equation. Set p , = h,/hkp, with h, taken as 1. Then, with a, = a, b, = 6 , and c k = c for all A, we have from (2.1.20) the relations

pk

Pk =

aqk-l

+ b,

q k = -c/pk

7

so that there results the difference equation

h, - bhk-1

+

achk-2 = 0.

A similar property holds for the periodic spline. 2.2. Existence, Uniqueness, and Best Approximation For most cases of interest, the proof of the existence of the spline function involves merely an application of Gershgorin’s theorem(cf.Todd [1962, p. 227]), which states that the eigenvalues of the matrix (ai,j) (i,j = 1, 2,..., n) lie in the union of the circles

1z

- aii

I

=

1 1 a i , j1

j#i

(1

< i < n)

in the complex plane. A matrix with dominant main diagonal (1 aii 1 > CiTi I ai,j 1) is nonsingular. I n (2.1.9) and (2.1.17), the sum 1 a i . j 1 is always equal to 1, with aii = 2. I n (2.1.8) and (2.1.16), the condition for dominant main diagonal is that A,, p, , po , A, be less than 2 in absolute value. Thus it is seen that the periodic cubic spline with prescribed ordinates at mesh points always exists and is unique, the representation being given by (2.1.2) with the M iuniquely determined by (2.1.9), and that the same is true in particular for nonperiodic splines having cantilevered ends (m, and m , prescribed), having simple end support ( M , = 0, M , = 0), having prescribed end moments, or having simple supports at points beyond the mesh extremities (e.g., Mo = AM, , M N = p M N - l , 0 < A < l , o < p < 1). A general existence theorem covering a much wider class of nonperiodic cubic splines is given in Section 2.9. There it will be necessary to prove special properties of the coefficient matrix. We remark that more than one spline may be associated with a set of values for the quantities M i. Replacing yi by yi + mxj + C for fixed m and C does not affect the right-hand member of (2.1.7). Boundary conditions (2.1.18) possess the same property. Thus, s,( Y ; x ) mx C =

+

+

2.2,

17

EXISTENCE, UNIQUENESS, AND BEST APPROXIMATION

+

+

Sd(P; x ) , where Ti= yi mxj C. For the periodic case, we may only say that S,(Y; x ) C = Sd(P;x), with pi = yj C. ( T h e only periodic linear function is a constant.) A related question concerns the arbitrariness of the quantities M ior m i . Is there always some spline associated with an arbitrarily prescribed set of values of M ior mi ? It is seen for periodic splines that adding the corresponding members of (2.1.5) ( j = 1, 2, ..., N ) gives the necessary condition

+

+

c (4 + h,+l)W N

?=I

= 0.

It may be seen, however, that any set of values of the M isatisfying this relationship is an admissible set. If we designate the left-hand member = 0. Set of (2.1.5) by $j, then the preceding equation implies (yl -YN)/h1 C. Then(yi -yj+l)/hj = c $1 ... $j-1 ( j = 2,..., N ) . $N , but T h e equation system requires (yl - yN)/hl = c $1 this is equal to c. We now have the relations

+ + +

x:l$i

+ + +

1

Y1 Y2

YN

+ hc, = Y1 + h2(c + $11, ... = + + + +

=YN

YN-1

hN(c

$1

' I '

4N-l).

These equations have a one-parameter family of solutions (parameter yN) ,[: hi($l ... $i-l)]/(a - b ) , which is a known iff we take c = C function of the given Mi's. It is readily seen for nonperiodic splines that there is no restriction upon the quantities M i. T h e corresponding problem for the slopes mi is somewhat more complex. If we designate by 3z,hj the left-hand member of (2.1.14) and set (yl - yN)/h12= c, Eqs. (2.1.14) ( j= 1,..., N ) require that the following equations be satisfied:

+ +

(Yz -Y1)lhz2 = 41 - c, (YB - Yz)/k2 = $2 - $1

...

(YN

-yN-l)lhN2

(Yl -yN)/h12

= #N-1 #N

+ c,

- #N-2 - #N-1

+ f (-l)N-z#l + (-l)N-lc, + ''. + (-l)"'h + (-l)Nc*

These are consistent iff z,hN - z,hNP1 This is equivalent to the condition N

i=l

ml

*'*

+ *.. + (-1)"-'

+ h,

mN

[ I - (-l)"]

z,hl

+ (-l)Nc

= c[l

-

= C.

(-1)N].

18

11.

THE CUBIC SPLINE

T h e resulting system of equations has a one-parameter family of solutions for (yl ,..., y N )iff in addition C[h12- hz2

+ + (--I)N-’hNZ]= -[hZ2#, + h3’(1CI2 - + **.

+

#1)

hN2(#N-l

- #N-2

**.

+ + ”*

(-1)N-2$1)]*

We restrict our attention here to the case in which the intervals are of equal length. If N is odd, these requirements are equivalent to #z .-* # N = 0 and c = $ N . If N is even, there $3 exists a two-parameter family of solutions y z j = y N+ h2($, + + $2j-l), ... $zj) (parameters y N and c) iff y2j+l= y N 15% hz($2 $zN-l = $, $2N = 0, that is, iff

+ + + + + + + + + + + +...+ m1

+ m3 + ... + mN-1

=

m2

+

+ m4 + .*. + mN = 0.

Equations (2.1.7) or, alternatively, (2.1.14), are given added significance if we consider the following extremal problem. Let f ”(x) be continuous. For a given mesh A , let fj = f(xj) and let Sd(f;x) denote the periodic spline of interpolation to f(x) or, alternatively, the nonperiodic spline satisfying end conditions (2.1.18i). Thus, Sd(f;xj) = fj . Let Sd(x) be any cubic spline on A . Form the integral E

=

Sb

[ f ” ( x ) - S:(x)lz dx.

a

T h e quantity E is, of course, a measure of the approximation of S,”(x) to f”(x) on [a, b]. Let Mj denote &’,”(xi). Expanding the integral and integrating by parts gives E the form

+ c h3. (Mj2_,+ Mj-lMj + M 2 ) ,

(2.2.1)

_1

j=1

wheref; = fh,fo=f N , M o=M,iff(x)andS,(x)haveperiodb-a.The function E has a stationary point in the nonperiodic case when the conditions

hj -3

(L+l- f j hii-1

+3hj+l M j + hy Mj+l -

fi

hj

) \1

-

0 (j = 1,2,...)N - I),

(2.2.2)

2.3.

CONVERGENCE

19

are satisfied. I n the periodic case, a stationary point exists when the second relation is valid for j = 1, 2, ..., N . These conditions are equivalent t o (2.1.8) with boundary conditions (2.1.18i) or to (2.1.9). Consequently, the function E has a stationary point among the various choices of M j iff Sd(x) = Sd(f ; x). We shall show that this stationary point is actually a minimum point. Denote by ( M o ,Ml,..., M N ) and (M1 ,..., M N ) the solutions of these equation systems for the nonperiodic and periodic cases, respectively; that is, Mi= Sl(f ; x,). We rearrange the expression for E as follows: multiply the algebraic expressions in Eqs. (2.2.2), with Mj in place of M i, by -2M0, -2M1 ,..., -2MN, respectively, and add to the right-hand side of (2.2.1). We obtain expressions for E in the forms

- M j B i P l - 2MjMi] =

=

j”[f”(x)]’ dx + a

h.

j=1

j”[ f “ ( x > l 2dx + j.” a

$ [(Mj-l - fli-d2+ ( M j - , - I@-l)(Mj - Mi)

a

[S;(x)

.)I2

- S;(f;

dx

-

Observe that the first and third terms in the last member are independent of the choice of M i .It is evident, therefore, that E is minimum for Si(x) = Si(f;x). This is the best approximation property of the spline interpolation. I n Chapter 111, this and other related extremal properties are explored by more elegant and powerful means.

2.3. Convergence The effectiveness of the spline in approximation can be explained to a considerable extent by its striking convergence properties. If f ( * ) ( x )is continuous on [a, b] ( q = 0, 1, 2, 3, or 4), we find that Sd(f;x) converges tof(x) on a sequence of meshes at least as rapidly as the approach to zero of the qth power of the mesh norm 11 A 11 = maxjhj. (To compare with the degree of convergence of more general approximating sequences, see Davis [1963, Chapter XIII].) Similarly, SLp)(f;x) converges to f ( ” ) ( x ) (0 p q) at least as rapidly as the ( q - p)th power of the

<


Using (2.1.5) and the interpolation property of SA(x), we obtain f y -M. 7

3 -

34

11.

THE CUBIC SPLINE

On [xi-1,xi], we evidently have f”(x) - S i ( X ) = f ; - M j

+

JZ

[f”’(t)-

dt,

2 7

and we obtain from this relation the inequality

G Q I1

II (3

+ Kz)CL(f”’;/I A ll),

where we have made use of (2.3.22). The corresponding properties for Sb(x)- f ’ ( x ) and Sd(x) -f(x) now follow by integration and the application of Rolle’s theorem as before. Two further properties serve to complete this initial presentation of spline convergence properties. We only mention these here and postpone their proofs to Sections 2.9 and 3.12. Iffiw(.) is continuous on [a, b] and if {A,} has the properties 11 Akll 0 < co, and maxi I A,,j - I + 0, then as k + co, /I A , ll/minjh,,j ---f


O and infk(4 - p k , N , ) > 0. Thus, 11 C-l 11 is uniformly bounded. We fotm the Nk x N k matrix 1 1

H=

-

+ ( 2 + A,)

[(4 - A,)

0

1 0

0

0 0

[o

0

A,] 0

0

0 0 0

"'

...

1

...

...

0

... 0

1 1 - [(4

-PN)~N-I

Then, if r represents the vector

we have C(u - Hr )

and the right-hand vector is

= (I -

CH)Y,

0' 0 0

+ (2 + PN)

0 PN-11

1.

72

11.

THE CUBIC SPLINE

T h e rate at which li(I - C H ) r 11 approaches zero is now evident, and the conclusions of the theorem follow from the boundedness of 11 C-l I/. Our final result concerns a rather curious property of convergence to the fi”(x). For the sake of simplicity, we restrict our attention to the periodic cubic spline, although the argument can be carried through for nonperiodic splines as well. We consider the jump in the spline third derivatives at xj and set

From the set of equations for the quantities ai ,

we obtain, by subtracting corresponding members of the j t h from the ( j 1)th equation and dividing the result by hj + hi+l+ the equation

+

+

DS

=g,

(2.9.22)

2.9.

ADDITIONAL EXISTENCE AND CONVERGENCE THEOREMS

73

and the columns by

We obtain as a result the matrix


; is a Cauchy sequence in L2(a,b). Since L 2 ( a ,b) is complete, we can find a function g ( x ) in L2(a,b) such that lim

N+m

b a

{g(x) - SJN(f;x)}, dx = 0.

(3.14.4)

Let (3.14.5)

then

by Schwarz’s inequality. Consequently, (3.14.4) implies that for each x in [a, b]

100

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

since lim I f ’ ( x ) - S i N ( f ;x)

N-tm

1

(3.14.7)

=0

uniformly for x in [a, b] by Theorem 3.8.1. Indeed, since

If’@)

-

G(x)

1

< If’(4- Si,(f; 4 1 + I G ( 4 - Si,(f; 4

I7

f ’(x) must be identical with G(x).This, however, implies that f “(x) = g(x) a.e., which establishes the theorem. Theorem 3.14.1 and Lemma 3.14.1 below provide the major tools needed to demonstrate the validity of the decompositions (3.13.3) and (3.1 3.4).

Lemma 3.14.1. Let A , C A , be two meshes on [a, b]. If SAl(x)and SA2(x)are splines on A , and A , , respectively, such that SA2(x)vanishes on A , , then (SA1, SAz)= 0 if SAl(x)is of type 11’, or SA,(x)is of type 1’, or both SAl(x)and SA2(x)are periodic. Proof. Let A , be defined by a = x,,< x, integrate (SAl, SA2)by parts twice, then




(3.14.8)

N=2

,X2(a, 6)

‘A,

@

W

N=2

@ [‘A,

- ‘AN-1]

‘

Am’

(3.14.10)

3.15.

101

CANONICAL MESH BASES A N D THEIR PROPERTIES

Proof. We prove only (3.14.8), since the proof of (3.14.9) and (3.14.10) are completely analogous. By definition, FA, is closed, and consequently we have the decomposition s 2 ( a ,b) = FA,

0GA, ,

(3.14.1 1)

where GAm denotes the orthogonal complement of FA,. Since FA C Z 2 ( a ,b), we need only prove that f(x) is in FA, if it is in Z 2 ( a ,b). Lg { S A N ( fx)} ; ( N = 1, 2, ...) be the sequence of type I splines of interpolation to f(x) determined by {A,) such that f(x) - SdN(f; x) is of type I’for each N . Then, by Theorem 3.14.1, lim

Nim

1l.f - sA,,f

11

= 0.

(3.14.12)

This proves the theorem, since FA, is closed, and Lemma 3.14.1 establishes the orthogonality of the component spaces.

REMARK3.14.1. I n establishing (3.14.9) and (3.14. lo), we choose the f;x)) to consist of type 11’ splines or periodic splines, sequence {SAN( respectively. I n (3.14.9), for each N , [F& -&’;,-,I is the family of type 11’ splines on A , whose defining values on ANpl vanish. In the periodic case we also use the fact that Z j ( a , b) is dense in Z 2 ( a ,6). We consider this in greater detail in Section 6.14.

3.15. Canonical Mesh Bases and Their Properties Let {A,} ( N = 1, 2, ...) be a sequence of meshes on [a, b] with A N C A,,, . We assume that we are given an orthonormal basis for FA,, and we extend this to an orthonormal basis for FA, by constructing an orthonormal basis for the orthogonal complement, [FA,-F,J, of FA,with respect to FA, . T h e construction yields for every N an orthonormal basis for [FA, - F A J .T h e same method yields with slight modifications orthonormal bases for [Fi, - FiJ and [PA, - PA,]. Theorem 3.14.2 then shows that this construction provides explicit orthonormal bases for X 2 [ a ,61 if I/ A , /I -+ 0 as N A 00. We call mesh bases these orthonormal bases for [FA, - FAJ, [F;, - F i J , and [PA,

- PAl].

Consider the set 42 of all distinct mesh points determined by the sequence of meshes { A N } excluding those that comprise A , . Since M is denumerable, let it have a specific enumeration M

= {PI , P ,

,...}.

(3.15.1)

102

111.

INTRINSIC PROPERTIES OF CUBIC SPLINES

I n the case where only a mesh basis for [FA, - FAJ is desired, M has only a finite number of elements and is denoted by M,. Let A,, be the mesh obtained by inserting the point P, into the mesh A , , and let p,(x) be the type I’ spline on A , , such that pl(P1) = 1 and pl(x) vanishes on A , . If Ali is the mesh obtained by inserting Pi into the mesh = A , ) , then pi(x) is the type I’ spline on A I i such that pi(Pi) = 1 and pi( x) vanishes on A l , i - l . Lemma 3.14.1 assures us that the sequence {pi(x)) ( i = 1, 2, ...) consists of mutually orthogonal type I’ splines and by the manner of its construction is a basis for [FA,,, - F A J , where Z S FAl

@ LFA1l

-FAl]

@ LFAlz

-FA1ll

@ *”

*

Suppose (3.15.2)

the resulting sequence { d i ( x ) )( i = I , 2, ...) of orthonormal type I’ splines is a basis for [FAl,m- FAJ. By proceeding in a similar manner, but requiring that p i ( x ) be of type 11’ or periodic, we are led to mesh bases - FiJ and [PA,,,- P A J , respectively, where the additional for [FL1,, definitions needed are obvious. These bases are not unique, since the construction depends on how M is enumerated; moreover, they are not the desired bases for [FA, - FAA, [ F l , - FiJ, and [ P A - PA,]. If, however, the process exhausts the points of A , for each N before any points not in A , are enumerated, then we obtain the desired mesh bases. We single out one natural way of making this type of enumeration and refer to the mesh bases generated by it as canonical mesh bases. T h e enumeration employed in their construction consists of ordering from left to right for each N the mesh points that are in A , but not in A , - , . Thus, when a mesh basis is canonical, its construction is completely defined and depends on a given sequence of meshes. Any mesh basis is canonical, however, with respect to the auxiliary sequence of meshes {A,,i} ( i = 0, 1,...) used in its construction.

Lemma 3,15,1, Let { d i ( x ) ) ( i = 1, 2,...) be a canonical mesh basisfor [FA, - FAJ, [F;, - F i J , or [ P A , - PAJ determined by a sequence of meshes (A,) ( N = 1, 2,...) with A , C A,,, . Let {Ali) (i = 1, 2,...) be the related sequence of meshes used in the construction of {di(x)). Then

I ,pyx)

j


/ 2

(5.9.3)

5.9.

167

CONVERGENCE OF LOWER-ORDER DERIVATIVES


- S,(f; x ) is of tYPeII', ( 4 f ( x ) and S d ( f ;x ) are periodic, then

Ilf

- S A ,Ila~ =

J-b (f a

- SA.,) . D2nfdx.

v.

170

POLYNOMIAL SPLINES OF ODD DEGREE

Theorem 5.10.2. Let A : a = xo < x1 < < x N = b and f ( x ) in X Z n ( a ,b) be given. If S,(f; x) is a polynomial spline of degree 2n - 1 and deJiciency k which interpolates to f ( x ) on A , satisjies the interpolation conditions . - a

S'm'(f;xi)

( a = 1, 2 ,..., k

=f'*'(~i)

-

1; i

=

1, 2 ,..., N - I), (5.10.7)

and one of the conditions, (a) f ( x ) - S,( f ; x) is of type 1', (b)f ( x ) - S,( f ; x) is of type 11', (c) f ( x ) and S,(f; x) are periodic, then

llf

-

S A ,/I2~ =

Jb

a

(f - S A , ~ Dznf ) dx. *

As we have already pointed out, the second integral relation generally is not valid for splines of type k. If we consider splines satisfying the slightly modified end conditions S y ( f; Xi) = f ' " ' ( X i ) (OL

= 0 , 1 , ..., k - 1;

OL

= n , n + 1,..., 2 n - k -

1; i = O , N ) ,

(5.10.8)

the second integral relation is again valid. I n this case, however, the minimum norm property fails, but the best approximation property holds. We refer to splines of interpolation of this form as splines of modijied type k . A type I1 spline is a spline of modified type I.

5.1 1. Raising the Order of Convergence I n this section, we proceed very much as in Section 3.10. From (5.9.3), it follows that If(,)(x) - Sp)(f;X)

1

< K,

1s

b

*

a

{f(")(x) - S y ) ( f ;x)}z dxt

1 I2

I/ d /l(zn-z,-1)/2

(5.11 .I) for a suitable choice of the constant K, . If the second integral relation is valid, (5.11.1) is equivalent to

I f'm)(4 - Sp)(f; 4 I

hence, we have

5.11.

171

RAISING THE ORDER OF CONVERGENCE

As in Section 3.10, we set a: = 0 and solve (5.11.2) for supz I f ( x ) - S4(f;x)I, which is possible except in the trivial case when f ( x ) = S,(f; x). Thus, S;P

).(fi

- Sd(f;

X)

1

< KO2

*

Vab[f'2n-1'] * 11 A

1 2n-1,

(5.11.3)

which, together with (5.11.2), implies I f ( , ) ( x ) - SSp)(f;X)

1

< K,

KO* Vab[f(2n-1)] * 11 A

(5.11.4)

l/2n--ol-1.

We are now able to reformulate Theorems 5.9.1 and 5.9.2.

Theorem 5.1 1.1.

Let f ( x ) be in A?2n(a, b), and let { A , : a = x o ~ < " ' < x ~ =b} h'

be a sequence of meshes with 11 A,I( -+ 0 as N + CO. Let {S+(f; x)} be a sequence of splines of interpolation to f(x) satisfying one of the conditions (a)f(x) - SAN(f; x) is of type I' ( N = 0, I,...), (b) f(x) and S+(f; x) are periodic ( N = 0, 1,...), (c) f ( x ) - S,(f; x) is of type II' ( N = 0, 1 ,...). Then (5.11.5.1) (a = 0, 1,..., n - I), f ' " ' ( ~ )= SSp'(f;X) T,(x) where

+

1 T,(X) 1

< n(n - 1)

(a

+ I)[@ - I ) ! ]

. Vab[Cf(2n-l)] , (1 A ,

*2-(2n-a-1)

/12n--ol-la

(5.1 1.5.2)

W h e n condition (c) is satisfied, the factor 2-(2n-a-1)must be omitted for x = a and x = b.

In reformulating Theorem 5.9.2, we employ splines of modified type

k rather than splines of type k.

Theorem 5.11.2, L e t f ( x ) be in X Z n ( a ,b), and let {A,

: a = xoN


, k. If n - 2k 2 < 1 in (.5.11.6.2), it is replaced by 1.

+

x

5.12. Convergence of Higher-Order Derivatives Let A : u = xo < < xN = b be given, and let f ( x ) be in Z Z n ( u ,b). I n addition, let S,,(f; x ) be a polynomial spline of degree 2n - 1 interpolating to f ( x ) on d such that

I f ( x ) - S,(f;x) 1

< Ko2

*

Yab[f'2"--1']

*

11 d p

-

1

(5.12.1)

is valid. For the equally spaced difference quotients 6;A,/[xi-1 , Xi]

where 6$A.,[xi-1, xi] xu,i with xi--l xu,$


1. Details of the proof of the final assertion of Theorem 5.12.2 can be found in Section 6.12.

Theorem 5.12.2.

Let f ( x ) be in 3Y2n(a,b) and (A, :a

= xoN

< ... < x," N

=

bf

be a sequence of meshes such that 11 A , 11 + 0 as N 3 co.Let R A N , defined by (H2.5.2), be bounded with respect to N , and let {SAN( f ;x)>be a sequence of polynomial splines of degree 2n - 1 which interpolate to f ( x ) on corresponding meshes A , and which satisfy one of the conditions (a) SAN(f;X ) is of modijied type k ( N = 1, 2, ...), (b) f ( x ) is in &';"(a, b) and sA,(f; x) is a periodic spline of type k ( N = 1,2,...). Then we have, uniformly f o r x in [a, bl, ( a = 0, I, ..., 2n - K). (5.12.7) f ( a ) ( x )= SF$(f;x) + O(ll A , l12n--or--k) Moreover, limN+coSyA(f;x)

=f

(a)(,)

uniformly in x for

01

= 0,

1,..., 2n - 2.

v. POLYNOMIAL

174

SPLINES OF O D D DEGREE

5.13. Limits on the Order of Convergence T h e discussion of limitations on the convergence of cubic splines contained in Section 3.12 carries over essentially unchanged to polynomial splines of odd degree. I n general, we have f ( a ) ( ~ )=

Spi(j;X)

+ O(1l

A N

1 212-a-1)

(a = 0,

1,...,2n - 1). (5.13.1)

For a = 0, I , ..., n - 1, no restrictions are imposed on the meshes, but I , ..., 2n - I , the mesh parameters RANmust be bounded for a = n, n as a function of N . For uniform spacing we obtained in the periodic case (cf. Section 4.6, Theorem 4.6.3) the stronger result that

+

pyx)

=

S'e' A , ( f ; x)

+ O(1i d,

( a = 0, 1,..., 2n - 1).

l / 2 n- a )

(5.13.2)

I n the case of (5.13.1), the rate of convergence is proportional to Vab[f(2n-1)], whereas for (5.13.2) the rate of convergence is proportional to IIf(2n)/ I m for f ( x ) in CZn(u,b). T h e rate of convergence in (5.13.2), insofar as it depends on 11 A , 11, cannot be improved. We have, in fact, the following theorem, the proof of which is the same as that of Theorem 3.12.1.

Theorem 5.13.1. Let {A,} be a sequence of meshes with 11 A N 11 -+ 0 as N 4 co,and let RANbe bounded with respect to N . Let f ( x ) be in C2n(u,b) and p > 0. If, uniformly f o ~x in [u, b],

f(.)

SLl,(f;

then

).

Dznf

+ O(l1

1/ 2 n+p) ,

(5.13.3)

= 0.

REMARK5.13.1. T h e splines involved in Theorem 5.13.1 need not be of deficiency one; the key hypothesis is (5.13.3). 5.14. Hilbert Space Interpretation We continue here the discussion begun in Section 3.13. T h e class

X n ( u , b), under the pseudo-inner product ( f ,g)

=

jbf(W . g ' n ' ( 4 dx a

(5.14.1)

is a Hilbert space if we identify functions that differ by a polynomial of g), degree n - I ; without these identifications and the inner product (f,

5.14.

HILBERT SPACE INTERPRETATION

175

A : a = x,, < x, < < xN = b be given, and let F,(n, k) denote the family of polynomial splines on A of n). As a linear subspace of degree 2n - 1 and deficiency k (k X n ( a , b), FA(n,k) has dimension k(N - 1) 2n, and as a Hilbert space, where splines differing by a polynomial of degree n - 1 are identified, it has dimension k ( N - 1) + n. If PA(n,k) denotes the family of periodic polynomial splines on A of degree 2n - 1 and deficiency k, then PA(n,k) is a subspace of FA(n,k ) . As a linear space (without any identifications), P,(n, k) has dimension N * k ; as a Hilbert space (with splines differing by a constant identified), PA(n,k) has dimension Nk - 1. If d, refines A , , FAl(n,k) is a subspace of Fdz(n,k), and PAl(n,k) is a subspace of P,,(n, k). Since they are finite dimensional, FA(n,k) and PA(n,k) are always closed subspaces. If A , C A , , then [FA,(=,k) - FA,(n,k)] and [PA,(n,k) - P J n , k)] denote the splines in FA2(n,k) or PAz(n,k), respectively, whose defining values (including any derivatives*) o n d , are zero. We employ similar notation for other spaces when required. T h e orthogonality of the component spaces in the decompositions in the remainder of this section is demonstrated in Section 5.15. Observe that the linear spaces [FAl(n,k) -FAi-l(n, k)],etc., unlike the spacesF,(n,k), are unaffected when systems differing by a polynomial of degree n - 1 are identified. Thus, we can regard elements in these spaces as functions rather than equivalence classes even after identifications are made. Consider now a sequence of meshes { A N } on [a, b] with A N C ( N = 1, 2, ...). I t is true that X n ( a , b ) is simply a linear space. Let


,(n,

4

c 0[Fi,(n, k) -FiN-Jn, 4 1 m

=F

p , k) 0

(5.14.6.1)

N--2

and T,,(n,k) as m

T,,(n> ')

=

Td,(n,

')

0

N-2

0

')

-

')I*

(5*14.6*2)

We establish in Section 5.15 that, for k # 0, T,,(n, k) = F i m (n,k) = %"(a, b),

if 11 A N 11

+0

(5.14.7)

as N + 00. 5.15. Convergence in Norm

Theorem 5.15.1. Let { A , : a = x o N < ... < , :x = b} be a sequence of meshes with A , C A,,, ( N = 1, 2,...), 11 d, 11 + 0 as N + co, let * A spline is of type k if for some function f ( x ) it is a spline of interpolation to f ( x ) of type k.

5.15.

CONVERGENCE IN NORM

177

f ( x ) be in %lz(a, b), and let {SA,(f;3)) be a sequence of polynomial splines of degree 2n - 1 and deficiency k (k a ) with S$;(f; xiN) = f(u)(xi”) (i = 1, 2,..., mN - 1; a = 0, 1,..., k - 1) f o r each N . If, in addition, one of the conditions, (a) f ( x ) - SdN(f; x) is of type I’ ( N = 1, 2, ...), (b) S , , ( f ; x ) i s o f t y p e I I ’ ( N = 1 , 2,...) , ( c ) S d N ( f ; x ) i s o f t y p e k ( N1=, 2,...), (d) f ( x ) and SdN(f; x) are in ,Xplz(a,b), ( N = 1,2,...), is satisfied, then Ilf - SdN.,ll + 0 as N + 00.




k)l'

+ 2 < 1 in (5.16.2.2), it is replaced by 1.

Proof. Except possibly at x = a and x = b, we can find points Zi such that Sin-l)(Zi) = 0 and I x - Zi1 &(n- 2k 2)1/2 11 11


0 such that m

S,(X) =

1 {Sp'(L;x ) } ~< /3=

(a = 0,

1,...,n

-

2).

(6.16.4)

i=l

T h e elements S,(L; x) of a canonical mesh basis not only have the property of orthogonality, but if d, C A,,, the basis for is simply an extension of the basis for FA,. If d, is defined by a = x,, < x1 < . - * < X, = b and if, to obtain a basis for FA,, we employ cardinal splines Si(d, ; x) whose defining values at all but the mesh point xi vanish and there only one defining value is nonzero, we not only lose orthogonality, but in passing from FAN to a completely new set of basis elements is needed. There is a definite analogy here between this situation and the use of Newtonian interpolation formulas rather than Lagrangian interpolation formulas. When additional interpolation points are added, a whole new set of Lagrangian unit functions is needed (Davis [1963, p. 411); on the other hand, the set of Newtonian functions can be supplemented to accomodate the new interpolation points.

6.17. Kernels and Integral Representations I n this section, we state the analogs for the case of generalized splines of the theorems contained in Section 5.17. T h e statement of the theorems, however, is such that heterogeneous splines are included. T h e proofs differ in no essential way from the earlier proofs and are consequently omitted.

Theorem 6.17.1. Let f(x) be in %&(a, b ) or Xpn(a, b), and let {Si(L,k ; x ) ) (i = 1, 2 ,...), together with {ui(L, k ; x)} (i = 1, 2 ,..., m), be a canonical mesh basis for Fdm(k,L),F i m ( kL, ) , Tdm(k, L ) , HAm(k,L ) , or PAm(k, L ) , which is an orthonormal basis for X " ( a , b) or Xpn(a,b). Then for every x in [a, b] H,,,(L, k; x, t ) Lf(t)dt

+ G(,)(x)

( a = 0, 1,

...,n - 1)

(6.17.1)

6.18.

APPROXIMATION OF LINEAR FUNCTIONALS

22 1

whereLG(x) = 0 for x in [a, b], Ha*,&, k; x, t )

c u y ( L , k; m

=

x)

*

Luz(L, k; t )

i=l

+ c S y L , k; x) N

LS,(L, k; t ) ,

(6.17.2)

i=l

and the limit exists uniformly with respect to x in [a, b].

Corollary 6.17.1. Let f ( x ) be in X 2 n ( a ,b) or Xp2"(a,b), and let {S,(L, k ; x)} (i = 1 , 2 ,...), together with {ui(L,k ; x ) } (i = 1, 2 ,..., m), be a canonical mesh basis (cf.footnote Section5.17) for FAw(k,L ) ,FAm(k,L), , is an orthonormal basis for T A m ( k , L ) ,H A m ( k , L ) ,or P A m ( k , L ) which X R ( a ,b) or .Xpn(a,b). Then for every x in [a, b] and a = 0, 1 , ..., 2n - 2, Eq. (6.1 7.1) is valid. I n the case of heterogeneous splines, the maximum deficiency is assumed not to exceed k . Theorem 6.17.2. Let (n-,} (i = 0, 1 , ...) be a sequence of meshes on [a, b] determining a canonical mesh basis forFAm(k,L ) ,Fim(k,L), TAm(k, L), Hdm(k,L),or PAm(k,L),which is an orthonormal basis for X n ( a , b) or Xpn(a,b).If H,,N(L, k;x, t ) is defined by (6.17.1.2), then for each X { H , , ~ ) (a = 0, 1 , ..., n - 2; N = 1 , 2 ,...) is a Cauchy sequence in L2(a,b) and, consequently, a Cauchy sequence in L(a, b). If H,(L, k ; X , t ) ( a = 0, l,,.., n - 2 ) denotes the common limit, then for f ( x ) in *%(a, b) or Xpn(a, b) (6.17.3) f ( , ) ( x ) = H J L , k; x, t ) * Lf(t)dt G("'(x), a

+

where L G ( x ) = 0 for x in [a, b]. Moreover, the convergence is uniform with respect to x in [a, b], and Ha(& k ; x, t ) is obtainedfrom H,-,(L, k ; x , t ) by formal term-by-term differentiation with respect to x.

REMARK6.17.1. We are tacitly assuming, when we have a canonical mesh basis for X n ( a , b) or Xpn(a,b), that 11 T, 11 = O ( l / i ) . 6.18. Representation and Approximation of Linear Functionals Analogs of the four theorems contained in Section 5.18 remain valid for generalized splines, and again the arguments needed to prove the theorems for generalized splines are essentially unchanged. Con-

222

VI.

GENERALIZED SPLINES

sequently, we again content ourselves with just the statement of the theorems and omit the proofs. We do, however, consider some examples of approximating linear functionals in which we approximate an integral using these equally spaced values of the integrand. These same examples are considered by Sard [1963, Chapter 111, and they illustrate the manner in which spline theory provides many of the "best approximations" obtained by Sard. We also connect generalized spline theory to the calculation of the eigenvalues of a linear differential operator. We conclude this section with an application of heterogeneous splines to the approximation of point functionals LZZ:f -+f(x).

Theorem 6.18.1. Let f ( x ) be in X a ( u , b) or -Xpn(a,b), and let {S,(k,L ; x)} (i = 1, 2 ,...), together with {u,(k,L ; x)} (i = 1, 2 ,..., m), be u canonical mesh basis f o r FAm(k, L ) , Fim(k,L ) , TAm(k, L), HAm(k, L ), or P,,(k,L), which is an orthonormal basis for -Xn(a, 6 ) or Xpa(a,b). If 9 is a lineur functional of the f o r m (.5.18.1), then HN(k,L ; t ) Lf(t)dt

+ Y o G,

(6.18.1 . l )

where

1 u?'(k,L; b

ai = 3-0

pi =

a

c j S?'(k,L; 'I

b

j-0

a

s) dpj(s)

(i = 0, 1 ,..., m), (6.18.1.3)

s) dpj(s)

(i = 1, 2,...,N ) , (6.18.1.4)

and LG(x) = 0 for every x in [a, b].

Corollary 6.18.1. Let f ( x ) be in X Z n ( a b) , or Xp(a,b), and let {S,(k,L ; x)} (i = 1, 2,...), together with {u,(k,L;x ) } (i = 1, 2,..., m), be a canonical mesh basis for FAm(k, L), Fim(k, L), TAm(k, L), HAm(k, L), or PAm(k,L), which is an orthonormal basis f o r %"(a, b) or ,Xp"(a, b). If 9is a linear functional of the f o r m 9 0

f

=

5 r f " ' ( t ) d&),

j=o

(6.18.2)

a

where each p j ( t ) is a function of bounded variation on [a, b], then (5.18.1) holds except that in this case 71 < 2n - k - 1.

6.18.

223

APPROXIMATION OF LINEAR FUNCTIONALS

REMARK6.18.1. I n both the theorem and the corollary, the function

G(x) is dependent on the function f ( x ) , but the kernels H,(k,L; z) are

not. T h e method of proof essentially depends on the uniform convergence of the spline sequence and its derivatives to f ( x ) and its derivatives, and not on the rate of convergence. Thus, in view of Theorem 6.12.3, we need only require 7 < 2n - 2 rather than 2n - k - 1 in Corollary

6.18.1.

Theorem 6.18.2. Let f ( x ) be in %"(a, b) or .&"(a, b), and let {S,(k,L ; x)} (i = 1, 2 ,...), together with {ui(k,L ; x)} (i = 1, 2,..., m), be a canonical mesh basis f o r FAm(K, L), Fim(k,L ) , TAm(K, L), HAm(k,L), or PAm(k, L ) , which is an orthonormal basis f o r X n ( a , b) or X p n ( a ,b). I f 9 is a linear functional of the f o r m (5.18.1) except that q n - 2, then {HN(k,L ; x ) } is a Cauchy sequence in L2(a,b) and, consequently, a Cauchy sequence in L(a, b). I f H(k, L ; x) denotes the common limit, then


N , and let {S,(f; t , s)} ( N = 1, 2, ...) be an associated sequence of splines of interpolation on these meshes to a functionf(t, s) in C24(9).Since, for M > N , SN(f;t, s) is a spline of interpolation to S,( f;t, s) on r N, it follows, if the first integral relation applies, that a = to"

Therefore, the sequence of real numbers

is monotonically increasing and is bounded above by

consequently, the sequence

is a Cauchy sequence in L2(%) and has a limit g(t, s) in L 2 ( 9 ) Proceeding . as in Chapter 111, we let

We then have

We can now conclude from Theorem 7.10.1, the definition of g(t, s), and Schwarz's inequality

7.13.

THE METHOD OF CARDINAL SPLINES

25 1

from which it follows that

Consequently, we have

<