Fractional Integrals and Derivatives Theory and Appl· ications
S.G. Samko A.A. Kilbas 0.1. Marichev
Fractional Integrals and Derivatives
FRACTIONAL INTEGRALS AND DERIVATIVES
Theory and Applications
Stefan G. Samko
Rostov State University, Russia Anatoly A. Kilbas
Belorussian State University, Minsk, Belarus Oleg I. Marichev
Belorussian StaJe Unive�sity, Minsk, Belarus
Gordon and Breach Science Publishers Switzerland
Australia
Bel gium
France Germany Great Britain
India Japan Malaysia Netherlands
Russia Singapore
USA
Copyright © 1993 by OPA (Amsterdam) B.V. All rights reserved. Published under license by Gordon and Breach Science Publishers S.A. Gordon and Breach Science Publishers
Y-Parc Chemin de la Sallaz 1400 Yverdon. Switzerland
Post Office Box 90 Reading, Berkshire RG I 8 JL Great Britain
Private Bag 8 Camberwell, Victoria 3124 Australia
3-14-9. Okubo Shinjuku-ku. Tokyo 169 Japan
58, rue Lhomond 75005 Paris France
Emmaplein 5 1075 A W Amsterdam Netherlands
Glinkastrasse 13- 15 0- 1086 Berlin Germany
820 Town Center Drive Langhorne, Pennsylvania 19047 United States of America
Originally published in Russian by Nauka i Tekhnika, Minsk in 1987 as
HHTerpany H UpOH3BO�H�e�6Horo UOPR�Ka H HeKOTOp�e HX UpHJIOZeHHR
© 1987 Nauka i Tekhnika, Minsk
Library of Congress Cataloging-in-Publication Data Samko, S.G. (Stefan Grigor'evich) [Integraly i proizvodnye drobnogo poriadka i nekotorye ikh prilozheniia. English] Fractional integrals and derivatives: theory and applications Stefan G. Samko, Anatoly A. Kilba'i, Oleg I. Marichev.
I
p. em. Includes bibliographical references and index. ISBN 2-8-8124-864-0 1. Integral equations.
2. Fractional integrals
A. (Anatolii Aleksandrovich) Ill. Title. QA431.S2413
1993
I. Kilbas, A.
II. Marichev, 0.1. (Oieg lgorevich) 93-26071
515'.43 -- dc20
CIP
No part of this book may be reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying and recording, or by any infonnation storage or retrieval system, without pennission in writing from the publisher. Printed in Singapore.
CONTENTS xv Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface to the English edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii Notation of the main forms of fractional integrals and derivatives xxv Brief historical outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � . . . . . . . . . . . . . . . xxvii ·- ·
.
.
Chapter 1 -- Fractional Integrals and Derivatives on an Inte rval............................................................. /
§ 1 . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The spaces H>. and H>.(p).................................. 1.2. The spaces Lp and Lp(p) . . . . . . . . . . �........................ 1.3. Some special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 1.4. �
§2. Riemann- Liouville Fractional Integrals and Derivatives . . . . 2.1. The Abel integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. 2.3. 2.4. 2.5. 2.6.
On the solvability of the Abel equation in the space of integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of fractional integrals and derivatives and their simplest properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractional integrals and derivatives of complex order . . . . . . . . Fractional integrals of some elementary functions . . . . . . . . . . . Fractional integration and differentiation as reciprocal operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition formulae. Connection with semigroups of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •
2.7.
1
1 1
7 14 23
28 29 30 33 38 40 43 46
vi
CONTENTS
§3. The Fractional Integrals of Holder and Summable Functions 3.1. Mapping properties in the space H A 3.2. Mapping properties in the space H$(p) . . . . . . . . . . . . . . . . . . . . . 3.3. Mapping properties in the space Lp . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Mapping properties in the space Lp(P ) . . . . . . . . . . . . . . . . . . . . . . • • • • • •
• • • •
• • • • •
• •
• • • • • • •
53 53 57
66
70
§4. Bibliographical Remarks and Additional Information to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Historical notes . .. ................................ ........ . 4.2. Survey of other results (relating to §§1-3) ...................
82 82 84
Chapter 2 - Fractional Integrals and Derivatives on the Real Axis and Half-Axis.......................................
93
§5. The Main Properties of Fractional Integrals and Derivatives 5.1. Definitions and elementary properties . .... . . . . . . . . . . . . . . . . . . 5.2. Fractional integrals of Holderian functions . ....... ..... . ... . 5.3. Fractional integrals of summable functions . . . . . . . . . . . . . . . . . . 5.4. The Marchaud fractional derivative . . . . . . . . . . . .. . . . . . . . . . . . . 5.5. The finite part of integrals due to Hadamard . . . . . . . . . . . . . . . 5.6. Properties of finite differences and Marchaud fractional derivatives of order cr > 1 .................................. 5.7. Connection with fractional power of operators . .. .. .........
93 93 98 102 109 112 116 120
§6. Representation of Function by Fractional Integrals of Lp-F\inctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.1. The space JCit ( Lp ) ............ ................ ......... .. ... 122 6.2. Inversion of fractional integrals of Lp-functions . . . , ... ...... 123 6.3. Characterization of the space Ia ( Lp ) .. . . . . . . . . . . . . . . . . . . . . . 127 6.4. Sufficiency conditions for the representability of functions by fractional integrals . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . . . . . 131 6.5. On the integral modulus of continuity of Ja(L,)-functions . . . 136 _
§ 7. Integral Transforms of Fractional Integrals and Derivatives 7.1. The Fourier transform . . . . . . . . . . . . . . . . . ... .. . . . . . . . . . . . . . . . . 7.2. The Laplace transform . .......................... . . ........ 7.3. The Mellin transform. . . . . . . . . . . . . . . . ... . ... . . . . . . . . . . . . . . . . § 8. Fractional Integrals and Derivatives of Generalized F\inctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .
8.1. 8.2.
Preliminary ideas . . . . . . . . .. . . . . . . . . . . . ..... . . . . . .. . . . . . . . . . The case of the axis R1• Lizorkin 's space of test functions . . .
137 137 140 142 145 14 5 146
CONTENTS
8.3. 8.4. 8.5. 8.6.
vii
154 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Schwartz's approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The case of the half-axis. The approach via the adjoint operator McBride's spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The, case of an interval .
§9. Bibliographical Remarks and Additional Information to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .- . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1. 9.2. 9. 3 .
160
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5-8 . . . . . . . . . . . . . . . . . . 163 172
Historical notes . Survey of other results (relating to §§ ) Tables of fractiopal integrals and derivatives . . . . . . . . . . . . . . . .
Chapter 3 - Further Properties of Fractional Integrals and Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 § 10. Compositions of Fractional Integrals and Derivatives with Weights . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1. 10.2. 10.3. 10.4 .
175
176 189 191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Compositions of two one-sided integrals with power weights . Compositions of two-sided integrals with power weights . . . . . Compositions of several integrals with power weights . . . . . . . . Compositions with exponential and power-exponential weights . . . . . ·-� . . .
.
§ 11. Connection between Fractional Integrals and the Singular Operator . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . ,. . . . . . . . . . . . . . . . . . . . . . . . .
199
The singular operator S . The case of the whole line . The case of an interval and a half-axis . . . . . . . . . . . . . . . . . . . . . . Some other composition relations . . . .
199 202 204 210
§ 12. Fractional Integrals of the Potential Type . . . . . . . . . . . . . . . . . . .
21 3
11.1. 11.2. 11.3 . 11.4.
. . . . . . . .. . . .. . . . . . . . . . . . . . . .. . . .. . . . .. . . . . . . . .. . . . . .. . . . . . . . . . . . . .. . .......................
The real axis. The Riesz and Feller potentials . . . . . . . . . . . . . . . On the "truncation" of the Riesz potential to the half-axis . . The case of the half-axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The case of a finite interval . .
214 218 221 222
§ 13. Functions Representable by Fractional Integrals on an Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
224
12.1. 12.2. 12.3. 12.4.
13.1. 13.2. 13.3. 13. 4.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .· .
224 . . . 229 234 238
The Marchaud fractional derivative oil an interval . Characterization of fractional integrals of functions in Lp Continuation, restriction and "sewing'' of fractional integrals Characterization of fractional integrals of Holderian functions .
·. . . . . . . .
CONTENTS
viii
13.5. 13.6.
Fractional integration in the union of weighted Holder spaces Fractional· integrals and derivatives of functions with a prescribed continuity modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
246 249
§ 14. Miscellaneous Results for Fractional Integra-differentiation of Functions of a Real Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
254
14.1. 14.2. 14.3.
14.4. 14.5. 14.6.
Lipschitz spaces n; and if;................................ Mapping properties of fractional integration in n;.......... Fractional integrals and derivatives of functions which are given on the whole line and belong to n; on every finite interval . Fractional derivatives of absolutely continuous functions . . . . The ffiesz mean value theorem and inequalities for fractional integrals and derivatives . Fractional integration and the summation of series and integrals ............... . . .. . . . . . ...... . . ... . . . . . .
254 256
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 . . . . . . . . . 275 ·.
§ 15. The Generalized Leibniz Rule . . . . . . . . . . . . . . . . .. .. . . . . . . . . . . . .
277
Fractional integro-differentiation of analytic functions on the real axis . 15.2. The generalized Leibniz rule . . . .
277 280
A sym pto tic
285
15.1.
.................................................. ............................ § 16. E an ons of Fractional Integrals . . . . . . . . . . . . . . 16.1. Definitions and properties of asymptotic expansions. . . . . . . . . 16.2. The case of a power asymptotic expansion . . . . . . . . . . . . . . . . . 16.3. The case of a power-logarithmic asymptotic expansion . . . . . . 16.4. The case of a power-exponential asymptotic expansion . . . . . . 16.5. The asymptotic solution of Abel's equation . . . '. . . . . . . . . . xp
si
.
. ...
§ 1 7.
Remarks and Additio na l Info rma tio n to 3 . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .
285 287 294 297 299
Biblio graph ica l Chapter
301
17.1. Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 17.2. Survey of other results (relating to §§ 10-16) . . . . . . . . . . . . . . . . 305
Chapter 4 Other Forms of Fractional Integrals and Derivatives........................................................ 321 -
§ 18.
Direct Mo difica tio ns
and Generalizations of Riem ann ............ ......... . . . . . . . . . . .
. Erdelyi-Kober-type operators. .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . Fractional integrals of a function by another function . . . . . . . Had amard fractional integro-d ifferentiation . . . . . . . . . . . . . . . . .
L io uville Fractio nal Integra ls
321
18.1. 18.2. 18.3.
322 325 329
CONTENTS
18.4. 18.5. 18.6.
ix
One-dimensional modification of Bessel fractional integra differentiation and the spaces H 6 ,p = L; . The Chen fractional integral . . Dzherbashyan 's generalized fractional integral . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 -344 § 19. Weyl Fractional Integrals and Derivatives of Periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . 347 19.1. Definitions. Connections with Fourier series . . . . . . . . . . . . . . . . 347 19.2. Elementary properties of Weyl fractional integrals . . . . . . . . . . 352 19.3. Other forms of fractional integration of periodic functions . . . 354 19.4. The coincidence of Weyl and Marchaud fractional derivatives 356 19.5. The representability of periodic functions by the Weyl fractional integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 19.6. Weyl fractional integration and differentiation in the space of Holderian functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 19.7. inWeyln fractional integrals and derivatives of periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 ; 19.8. The Bernstein inequality for fractional integrals of trigonometric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 .
§ 20. An Approach to Fractional Integra-differentiation via Fractional Differences (The Griinwald-Letnikov Approach)
371
371 . . . . . . . . . . . . . . . . . . . 376 382 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 §21. Operators with Power-Logarithmic Kernels . . . . . . . . . . . . . . . . . 388 21.1. Mapping properties in the space H>......................... 389 21.2. Mapping properties in the space H6(P)..................... 396 21.3 . Mapping properties in the space Lp......................... 401 21.4. Mapping properties in the space Lp(p)...................... 404 21.5. Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 §22. Fractional Integrals and Derivatives in the Complex Plane 414 22.1. differentiation Definitions and the main properties of fractional integra in the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . 416 22.2. Fractional integra-differentiation of analytic functions . . . . . . . 420 22.3. functions Generalization of fractional integra-differentiation of analytic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 20.1. 20.2. 20.3. 20.4.
Differences of a fractional order and their properties . . . . . . . . Coincidence of the Griinwald-Letnikov derivative with the Marchaud derivative. The periodic case . . Coincidence of the Griinwald-Letnikov derivative with the Marchaud derivative. The non-periodic case . . . . . . . . . . . . . . . . Griinwald-Letnikov fractional differentiation on a finite interval .
CONTENTS
X
§23. Bibliographical Remarks and Additional Information to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
431
23.1. Historical notes . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 431 23.2. Survey of other results (relating to §§ 18-22) . . . . . . . . . . . . . ... 436 23.3. Fractional Answers to some questions put at the Conference on Calculus ( New Haven, 1974) . . . . . . . . . . . . .. . . . . . . . 455
C h ap t e r 5 Fract ional Int egro-differentiat i o n of Functions of Many Varia bles ................................. 457 -
§24. Partial and Mixed Integrals and Derivatives of Fractional Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24.1. The multidimensional Abel integral equation . . . . . . . . . . . . . . . 24.2. Partial and mixed fractional integrals and derivatives . . . . . . . 24.3. The case of two variables. Tensor product of operators . . . . . . 24.4. spaces Mapping properties of fractional integration operators in the L;;(Rn) ( with mixed norm) . . . . . . . . . . . . . . . . . . . . . . . . . . 24.5. Connection with a singular integral . . .... . . ... . . . . ..... . .. . . 24.6. form Partial and mixed fractional derivatives in the Marchaud ....................................................... 24.7. Characterization of fractional integrals of functions in L;;(R2) 24.8. Integral transform of fractional integrals and derivatives . . . . 24.9. Lizorkin function space invariant relative to fractional integrodifferentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.10. Fractional derivatives and integrals of periodic functions of ma.Ily variables . ..... .. . . . . . . . . . .. .. .... . . . . . ... . ... . . . . .... 24.11. Griinwald-Letnikov fractional differentiation . . . . . . . . . . . . . . . . 24.12. Operators of the polypotential type . . . . . . . . . . . .. . . . . . . . . . . . . t ro iati n . . . . . . . . . . . . . . . . . . . . . 25.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2. The Riesz potential and its Fourier transform. Invariant Lizorkin space . . . . . .. . . . . ... . .. . . .. . .. . . . . . ... . . . ... . . . . . . . . 25.3. and Mapping properties of the operator Ja. in the spaces L11(Jl!l) L11(Rn;p).............................................. 25.4. Riesz differentiation (hypersingular integrals) . .............. 25.5. Unilateral Riesz potentials . . ............... . ................
§25. Riesz Fra ctio na l In eg
-
diff er ent
o
.
§26. Hy persingular Integra l s a nd th e Spa ce of Riesz Po tentia l s
458
458 459 463 464 466 468 471 473 475 476 479 480
483
484 489 494 498 502
505
26.1. Investigation of the normalizing constants dn,l(cr) functions of the parameter cr . ......... ................. ..... .. ....... 505 as
CONTENTS
26.2. Convergence of the hypersingular integral for smooth func tions and diminution of order l to l > 2[a/2) in the case of a non-centered difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3. The hypersingular integral as an inverse of a Riesz potential 26.4. Hypersingular integrals with homogeneous characteristics . . . 26.5. Hypersingular integral with a homogeneous characteristic as a convolution with the distribution . . . . . . . . . . . . . . . . . . . . . . . . . 26.6. Representation of differential operators in partial derivatives by hypersingular integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.7. The space px ( Lp) of Riesz potentials and its characterization in terms of hypersingular integrals. The space L;,r ( Rn ) . . . . .
xi
510 512 518 525 527 532
§ 27. Bessel Fractional Integra-differentiation . . . . . . . . . . . . . . . . . . . . .
538
27 . 1 . The Bessel kernel and its properties . . . . . . . . . . . . . . . . . . . . . . . . 27.2. Connections with Poisson, G auss-Weierstrass and metaharmonic continuation semigroups . . . . . . . . . . . . . . . . . . . . 27.3. The space of Bessel potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.4. The realization of (E-A)a/2, a > 0, in terms ofhypersingular integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
538
§28. Other Forms of Multidimensional Fractional Integradifferentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28. 1. Riesz potential with Lorentz distance ( hyperbolic Riesz potentials ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28.2. Parabolic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 28.3. The realization of the fractional powers (-A�+ gt) a / and 2 (E- A�+ gt)a/ , a > 0, in terms of a hypersingular integral 28.4. Pyramidal analogues of mixed fractional integrals and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
§29. Bibliographical Remarks and Additional Information to Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29.1 . Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2. Survey of other results ( relating to §§ 24-28) . . . . . . . . . . . . . . . .
541 543 547 554
555 562 565 569 580
580 584
Chapter 6 - Applications to Integral Equations of the First Kind with Power and Power�Logarithmic Kernels 605 § 30� The Generalized Abel Integral Equation . .. . . . . . ......... ... 30. 1. The dominant singular integral equation . . . . . . . . . . . . . . . . . . . .
30.2. The generalized Abel equation on the whole axis . . . . . . . . . . . . 30.3. The generalized Abel equation on an interval . . . . . . . . . . . . . . . 30.4. The case of constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .
606
606 610 6 16 622
CONTENTS
xii
§31. The Noether Nature of the Equation of the First Kind with Power-Type Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31.1. 3 1.2. 3 1 .3. 3 1.4. §32.
629
Preliminaries on Noether operators . . . . . . . . . . . . . . . . . . . . . . . . . The equation on the axis ................................... Equations on a finite On he stability of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
630 634 646 657
Equations with Power-Logarithmic Kernels . . . . . . . . . . . . . . . . .
659
32. 1. Special Volterra functions and some of their properties . . . . . . 32.2. The solution of equations with integer non-negative powers of logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3. The solution of equations with real powers of logarithms . . . .
661
interval . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
t
s
§ 33. The Noether Nature of Equations of the F ir t
Kind
with
Po wer- Lo ga rithm ic Kernels. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .
33. 1. 33.2.
Im dding theorems for the ranges of the operators 1:/ and . . . . . . . . ·. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connection between the operators with power-logarithmic kernels and singular operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
be I�! .
.... . . . .
.
33.3. The Noether nature of equation (33.1) . . . . . . . . . . . . . . . . . . . . . . §34. Biblio graph ica l
664 667 672
673 674 681
Remarks and Additional Information to
Chapter 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
684
34. 1. Historical 34.2.
684 687
notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Survey of other results (relating to §§ 30-33) . . . . . . . . . . . . . . . .
Chapter 7 Integral Equations of the First Kind with Special Functions as Kernels.................................. -
§35. Som e Equa tio ns Ga uss
and
with
Homo geneo us Kernels Invo lving
L egendre Functio ns. . . . . . . . . . . . .. . . . .. . . . . . . . . . . . .
Equations with the Gauss function . . . . . . . . . . . . . . . . . . . . . . . . 35.2. Equations with the Legendre function . . . . . . . . . . . . . . . . . . . . . .
35. 1.
.
§36. Fractio na l Integra ls a nd Derivatives as Integra l Tra nsforms
De fintheir itioncharacterization of the G-transform.. . .The. . spaces. . .rot;,;(L) and L�e,..,) and .................. 36.2. Existence, representations of the G-transform:ma. .ppi. . .n.g. . pr. . .op. .er. .t.ie. s. . .and ........................... 36.3. Factorization of the G-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4. Inversion of the G-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.5. The mapping properties, factorization and inversion of fractional integrals in the spaces rot;,;(L) and L�e,..,) . . . . . . .
36. 1.
695
. .
. . ...
�
.
696
696 699 703
704 709 7 13 7 16 720
CONTENTS
xiii
36.6. Other examples of factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.7. Mapping properties of the G-transform on fractional integrals and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.8. Index laws for fractional integrals and derivatives . . . . . . . . . . .
722
§ 37. Equations with Non-Homogeneous Kernels . . . . . . . . . . . . ... ..
730
37. 1 . Equations with difference kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.2. Generalized operators of Hankel and Erdelyi-Kober transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.3. Non-convolution operators with Bessel functions in kernels . . 37 .4. Equation of compositional type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.5. The W-transform and its inversion . . . . . . . . . . . . . . . . . . . . . . . . . 37.6. Application of fractional integrals to the inversion of the W-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
726 727 731 737 741 746 752 758
§ 38. Applications of Fractional Integra-differentiation to the Investigation of Dual Integral Equations . . . . . . . . . . . . . . . . . . . .
76 1
38.1 . Dual Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.2. Triple equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
762 768
§ 39. Bibliographical Remarks and Additional Information to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
772
39.1 . Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.2. Survey of other results (relating to §§ 35-38) . . . . . . . . . . . . . . . .
Chapter
8
772 775
- Applications to Differential Equations . . . . . . . 795
§40. Integral Representations for Solution of Partial Differential Equations of the Second Order via Analytic Functions and Their Applications to Boundary Value Problems . . . . . . . . . . .
795
40. 1 . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2. The representation of solutions of generalized Helmholtz two-axially symmetric equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3. Boundary value problems for the generalized Helmholtz twoaxially symmetric equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
796
§41. Euler-Poisson-Darboux Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
812
4 1 . 1 . Representations for solutions of the Euler-Poisson-Darboux equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2. Classical and generalized solutions of the Cauchy problem . . 41.3. The half-homogeneous Cauchy problem in multidimensional half-space . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4. The weighted Dirichlet and Neumann problems in a half-plane
800 809
813 819 823 826
xiv
CONTENTS
§42. Ordinary Differential Equations of Fractional Order . . . . . . .
42. 1. The Cauchy-type problem for differential equations and systems of differential equations of fractional order of general form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. The Cauchy-:type problem for linear differential equation of fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3. The Dirichlet-type problem for linear differential equation of fractional order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4. Solution of the linear differential equation of fractional order with constant coefficients in the space of generalized functions 42.5. The application of fractional differentiation to differential equations of integer order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
829
830 837 843 846 849
§43. Bibliographical Remarks and Additional Information to Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .- . . . . . . . . . . . . . . . . . . . . . . . .
856
43. 1. Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2. Survey of other results ( relating to §§ 40-42) . . . . . . . . . . . . . . . .
856 858
Bibliography .. . .. �...................................................... Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
873 953 96 5 973
FOREWORD The concepts of fractional differentiation and integration are usually associated with the name of Liouville. However, the creators of differential and integral calculus had already considered derivatives not only of integer order, but of fractional order too. On reading this book we learn that fractional derivatives were the subjects of Leibniz's study. Euler also took an interest in fractional derivat ives. Liouville, Abel, lliemann, Letnikov, Weyl, Hadamard and many other well-known mathematicians of the past and present influenced the development of fractional integra-differentiation, whic h has now become a significant topic in mathematical analysis. Integrals and derivatives of integer order are the normal integrals and derivatives of analysis. But in the case of fractional order these ideas manifest their own peculiar features. Because various modifications arise naturally in different situations, the interconnections between these modifications need to be investigated. Fractional derivatives and integrals have many uses and they themselves have arisen from certain requirements in applications. Although there are many individual research papers on fractional derivatives and integrals, a unifying monograph on the topic has never been published. The present book fills this gap. It is written by a prominent specialist in mathematical analysis, Prof. Dr. S.G. Samko (Rostov State University), together with Dr. A.A. Kilbas and Dr. 0.1. Marichev (Belorussian State University) . This monograph is an extensive, yet compact, exposition on the present state of mathematical research on fractional integra-differentiation. The authors themselves have made valuable contributions to the theory of fractional integration and differentiation and so it is natural that their own results take up a rather conspicuous amount of space in this book. The book is constructed in the following way. The main sections in each chapter acquaint the reader with the fundamental questions. The general theorems are proved in full as a rule ; although the reader is sometimes referred to original sources. All chapters include a . historical resume with results and sources. Many statements which supplement the fundamental text are given here without proof.
xvi
FOREWORD
Part of the text is devoted to the case of one variable, and the rest to several variables. The multidimensional case is especially interesting. Only in special cases does it reduce to combining the known one-dimensional results. In the case of many variables such sub jects as the fractional integro-differentiation, theory of Riesz, hypersingular integrals, Bessel fractional integro-differentiation, fractional power of hyperbolic, parabolic differential operators and others are considered. This book also contains a chapter devoted to integral equations with power or logarithmic-power kernels. Here the integro-differential operators already discussed are applied to solve rather general integral equations. In the process, the necessity of using the classical results of Muskhelishvili and Gahov become apparent. The authors of this monograph, who come from the Gahov school, are masters in these methods and have had much to do with their development. As well as this the book also contains a great deal of theory on integral equations of the first kind with special functions in the kernel, the solutions of which are obtained by means of fractional integro-differentiation. In the final chapter applications to some problems in differential equations are given. The presentation of t he monograph uses simple everyday language based on the knowledge of differential and integral calculus, usually well within the limits of courses taught in physics, mathematics and engineering faculties. This makes the book easily understandable to a wide circle of readers. The book will be of interest to anyone interested in mathematical analysis. It may serve as an introduction to questions connected with the idea of fractional integration and differentiation. There is no doubt that the book will be useful to specialists both as a reference book with its large bibliography and as a subject of study. I believe that the monograph will be a success, and I wish it good luck. Academician S.M. Nikol'skii Steklov Institute of Mathematics, Moscow
PREFACE TO THE ENGLISH EDITION This book contains not only a logical presentation of the main principles of fractional calculus, but also surveys the numerous special investigations involving fractional integrar-differentiation. Thus, while translating the book into English we were unable to avoid adding to these surveys papers which appeared after the Russian edition of the book was published in 1987. This was not an easy task: fractional calculus has not ceased in its development. We may refer, for example, to the 3rd Conference on Fractional Calculus, held in Tokyo, 1989 (see its proceedings in "Fractional Calculus and Applications" , Ed. K. Nishimoto, Nihon Univ ., Japan, 1990) and to the fact that about four hundred new references appear in the Bibliography of the English translation. Thus the English edition is rather expanded mainly due to the reviewing of §§ 4, 9, 17, 23, 29, 34, 39 and 43. Most additions were made to §§ 23 andc 29, and certain parts of § 43 were rewritten. All the surveys, including the ad�itions, are theoretical in character. We are not concerned in this book with the applied aspects of fractional analysis, such as in engineering, modelling, mechanics and so on. We draw your attention, however, to the papers by Simak [1) ( 1987), Bagley [1) (1990) and the book by Gorenflo and Vessela [3) ( 1991) which appeared after the Russian edition was published. Large bibliographies of applied investigation may by found in these publications. Our method of citing references is to emphasize the year of publication in the historical commentaries (§§ 4. 1, 9.1, 17. 1 , 23.1, 29.1 , 34.1, 39.1, 43.1) . In the main body of the text and in the reviews (§§ 4.2, 9.2, 17.2, 23.2, 29.2, 34.2, 39.2, 43.2) the year of publication is not indicated, except perhaps in the few cases when it may be important. There are also some slight alterations in the main text of the book. They were made either in connection with new information or to improve the presentation. The Russian text was rewritten in English by S.G. Samko (Preface, Brief Historical Outline, Foreword, §§ 2, 4-9, 12- 14, 17-20, 22-31) and A.A. Kilbas (the remaining sections) and we hope that the reader will appreciate the extent of this great achievement.
xviii
PREFACE TO THE ENGLISH EDITION
We would like to emphasize that our interests are mainly in the field of real analysis. That is why fractional calculus in the complex plane, although considered in § 22, plays second fiddle in this book in comparison with fractional analysis of one and many real variables. Nevertheless, we present a fairly comprehensive survey of investigations within the framework of complex analysis in § 23. We would like to express our thanks to Prof. R. Gorenflo for helpful information on some recent publicat ions. Last but not least, our deep thanks are owed to Galina Smirnova, Yulia Zhdanova, Tatyana Bessonova and Igor Tarasyuk for the patient and careful typing of this manuscript.
PREFACE The field of mathematical analysis entitled fractional calculus, which deals with the investigation and applications of derivatives and integrals of arbitrary (real or complex) order has a long history described in the Brief Historical Outline. It is a complex topic having interconnections with various problems of function theory, integral and differential equations, and other branches of analysis. It has been continually developed, stimulated by ideas and results in various fields of mathematical analysis. Fractional calculus of functions of one and many variables continues to be developed intensively. This is demonstrated both by the many publications - hundreds of papers in the past years - and by the international conferences devoted to the problems of fractional calculus. The first such conference was held in 1974 (New Haven, USA ; Proceedings in "Fractional calculus and its applications" , Ed. B. Ross, Lect. Notes Math., 1975, v. 457) , the second - in 1984 (Glasgow, Great Britain; Proceedings in "Fractional calculus" , Eds. A.C. McBride and G.F. Roach, Res. Notes Math., 1985, v. 138). Considering the long history of the develop ment of fractional analysis it is a surprising fact that few if any monographs devoted to this topic have appeared. Indeed world mathematical literature can not point to any book which would thoroughly and comprehensively reflect the achievements of this theory. The only book specially devoted to fractional calculus by Oldham and Spanier [1] ( 1974) was written by specialists in the applied problems of chemistry, and contained only a presentation of some class ical points of the theory. The main attention in the book was focused on the evaluation of fractional integrals and derivatives of concrete functions, and to ap plications to diffusion problems. Books which contain a chapter or a section concerning certain questions in the field of fractional calculus are mentioned, for example, Zygmund [6], Dzherbashyan [2], Sneddon [3], [6] , Butzer and Nessel [1] , Butzer an d Trebels [2], Davis [3], [4], Okikiolu [7], Samko [31], Fenyo an d Stolle [1]. The publication of the little known thesis of Marke [1] ( 1942), in Danish, by Copenhagen University, is also of certain interest for specialists. Lastly we have singled out papers which contain historical outlines of the development of fractional calculus. The first such outline appeared in the paper by Letnikov [2] ( 1868). There are also historical outlines in the papers by Davis [3] , [4] , Mikolas [6], Ross
_
XX
PREFACE
[1]-[3] , Tremblay [1, p. 12-19] . In the main they are devoted to the classical period of the development of fractional calculus. Perhaps the reasons for the absence of a unifying monograph on fractional calculus was because of the very rapid development of the theory of fractional integro-differentiation in t he last decades, and also its multifarious branching especially in the case of many variables. The absence of such a monograph in a way became a hindrance to the development of fractional calculus. Some results, amongst which were those of a fundamental and essential nature, were published in original papers, several of which were difficult to find and were little known. This inevitably created a situation where investigators wasted much effort obtaining results already known or readily derived from known ones. Also some papers contained mistakes caused by the incorrect interpretation of the basic ideas of the theory. Indeed the history of fractional calculus is replete with many papers where results already known were rediscovered, sometimes by the same methods that predecessors had used, and sometimes by quite other means. This situation was aggravated by the existence of many different approaches to fractional integr o di fferentiation, and consequently by many different fields in fractional calculus. Comparison of these approaches was seldom carried out and was comparatively little known. A researcher starting in the field often encountered inconvenience caused by the necessity to orientate himself or herself in the many diverse definitions of fractional integro-differentiation and in the enormous flow of publications. The authors of this book have their interests in the theory of integral operators, function theory, integral and differential equations and special functions, and they have used the apparatus of fractional integro-differentiation in their investigations since 1967. In the authors' investigations the necessity of obtaining results in the theory of fractional integro-differentiation arose frequently, and gradually the interests of at least the first of the authors shifted to the fractional calculus - firstly to functions of a single variable, and since 1974 to functions of many variables. In their work the authors gradually arrived at the idea of writing a book which would reflect the modern state of fractional calculus, and present its applications to the theory of integral operators and integral and differential equations. The wide bibliographical search undertaken by the authors and the analysis of the enormous number of papers strengthened the authors' idea. An essential role was also played by the fact that since 1968 the first author had given lectures on fractional integro-differentiation of functions of one and many variables to undergraduates and postgraduates of Rostov State University. The temptation to present all the important re sults of the theory known up to now with complete proofs was great. However, such an approach would require a multi-volumed edition. Thus, the authors found it more expedient to single out from the main text the distinctive historical-surveying sections which complete every chapter. These sections (§§ 4, 9, 17, 23, 29, 34, 39, 43) p rovide historical commentaries to the content of the preceding chapter, and contain the discussion and formulation of results which are close to the subject matter of the chapter, but were not included in the main text. These commentaries and results are divided
PREFACE
xxi
into paragraphs, whose enumeration are re el vant to the corresponding section. For example, § 4.1 contains historical information relevant to Chapter 1 and consists of paragraphs giving information on each of the §§ 2. 1-3.2 of the chapter. The second subsection, 4.2, presents a survey of results on the subject of Chapter 1 and consists of § § § 2.1-2.7 and 3. 1.-3.4. The authors were faced with the difficult problem of selecting material for the main text, and the authors' tastes natur ally influenced this choice. The resu lts presented in the main body of the chapters are, as a ru le, given with the complete proof. The first five chapters of the book contain the presentation of the theory of fractional integro-differentiation itself. Chapt ers 1-4 deal with functions of one variable and Chapter 5 with functions of many variables. Chapters 6-8 contain applications to integral and differential equations. The application of fractional integro-differentiation to multidimensional integral equations is not discussed in this book. Such applications may be found in the book by Samko [31) and in the review paper by Samko and Umarkhadzhiev [3). We call the reader's attention to § 23.3, where some questions which were p osed at the 1st Conference on the Fractional Calculus (New Haven, 1974) were answ ered and § 9.3, which presents tab les of fractional integrals and derivatives of some elementary and special functions. There are a large number of references given in this book covering a great number of publications describing both theory and applications. The reader will find references to many papers which may prove to be new both to specialists and historians in fractional calculus. The authors do not concern themselves with the theory of fractional powerof operators, as this would lead the text too far astray, although this topic is touched upon episodically as, for example, in § 5.7. The book also deals with the symbolical calculus of Boole and Heavyside, and does not the ideas of G- and H -functions of many variables, as the theory of these functions i s only in the first stage of development. The sign ificance of fractionalintegrals and derivatives or of Abel-type equations in application must especially be emphasized. This mathematical apparatus is used in various sciences such as physics, mechanics, chemistry and others. After the known Abel problem on the tautochrone (Abel [1) (1823)) the first applications were made by Liouville [1) (1832) to problems of geometry, physics and mechanics. Amongst them we may find the Laplace problem concern ing the influence of an ·in finite rect il inear conductor on a magnet, the Amp ere problem of the interaction of two such conductors, prob lems connected with attraction of bodies, the prob lem of the heat distribution in a ball, the Gauss problem of approximate quadratures and others. The survey of the applied problems considered by Liouville in the paper by Letnikov [4] ( 1874), 21-44, is worth seeing in this connection. There are many papers of purely applied character which use the methods of fractional c alculus, but this book does not deal with applications in other fields except mathematics. The applications of fractional calculus to integral and differential equations presented in Chapters 6-8 are of theoretical mathematical character themselves. The readerinterested in purely applied aspects of fractional use
PREFACE
xxii
calculus should refer to the following publications: Oldham and Spanier [1], mentioned above, which contains the chapter "Application to diffusion problems" ; the paper [2] by the same authors which contains a large list of papers with applications to chemical physics, hydrology, random processes, viscoelasticity, gravitation theory and so on; the book "Abel inversion and its generalization" ( Novosibirsk, 1978) and, in particular, the introductory paper by Preobrazhenskii [1] in this book; and the Proceedings of the 1st Conference on the Fractional Calculus, mentioned above. Some other publications are of relevance as well: the books by Tseitlin [1] {1984) in particular, pp. 275-276, Yu.I. Babenko [1] {1986) and the papers by Brenke [1] (1922), ROthe (1] {1931), Rabotnov (1] {1948), Bykov and Botashev [1] { 1965), Shermergor {1966), Fedosov [1] {1978), Gomes and Pestana [1] {1978), Z aganescu (1], (2] {1982), Bagley and Torvik [1] (1986), Koeller [1] {1986), Gorenflo and Vessela [2] {1986). Finally note that the term "fractional" integra-differentiation is used throughout this book. Sometimes this word arouses objections since the order of ''fractional" integra-differentiation is an arbitrary number, not necessarily a fractional one. However, the authors consider it inexp edient to change this historically established term. The authors hope that they have succeeded in presenting the various approaches to fractional integra-differentiation, and in acquainting the reader with the interconnections between the m and in clarifying the question about the complete coincidence of some of these approaches, including the coincidence of the domains of definition. Sections§§ 2, 4-6, 8,9, 2-14, 17-20 (except §18.1) 22-31, and the Brief Historical Outline were written by S.G. Samko. Sections §§ 15, 16, 21, 28.4, 32 and 33 were written by A.A. Kilbas. Sections § § 7 {except § 7.1), 10, 35-38, 40-42 were writt en by 0.1. Marichev. Sections § § 3, 11, 34 were cowritten by Samko and Kilbas, §§ 39, 43 and 18.1 by Kilbas and Marichev and § 1 by Samko, Kilbas and Marichev. The authors note that B.S. Rubin read a considerable part of the manuscript and made a number of valuable suggestions. This was also done by Vu Kim Than with respect to some sections. Some materials prepared by N .A. Virchenko were used in § 38, by Vu Kim Than - in § 36, by S.B. Yakubovich - in § 37.5 and § 37.6 and by V.S. Adamchik and A.V. Didenko - in § 42. Qualified assistance in the preparation of the manuscript was also given by V.A. Nogin and B.G. Vakulov. Useful information and assistance in finding a number of papers were rendered by R.G. Buschman, LB. Dimovski, B. Fisher, H.-J . Glaeske, R. Johnson, S.L. Kalla, K.S. Kolbig, E.R. Love, A.C. McBride, M. Mikolas, B. Muckenhoupt, K. Nishimoto, S. Owa, B. Ross, M. Saigo, R. Wheeden. The authors express their gratitude to all of them.
1
INTRODUCTION The subject of this book is differentiation D 01 and integration /01 of arbitrary order and some of their applications to integral and differential equations. Most of the theory of such operations is concerned with functions of one variable, but Chapter 5 is concerned with different forms of fractional integro-differentiation of functions of many variables. The discussion is presented mainly for functions of a real variable, but § 22 is devoted to functions of a complex variable. In order to make the book suitable for as wide a readership as possible we start the discussion from simpler properties and statements and pass from special cases to general ones. For this reason we deliberately investigate "model" cases before considering the most general problem. In many cases, especially towards the beginning of the book, we prefer proposition with simpler forms and proofs. More complicated cases are dealt with in later chapters or in final sections of the appropriate chapters. The characteristic feature of the book is an exposition of practically all known forms of fractional integro-differentiation and their comparison with each other. In many cases not only the identical nature of different forms to each other in certain spaces of functions is proved, but the coincidence of their domains of definition is also shown. Another distinguishing feature of the book is that we stress the problem of the representability of a function l(z) by the fractional integral I = l01cp, a > 0, of a function cp belonging to one or another given space X . This question is investigated in all situations considered in the book - for functions of one and many variables, in periodic and non-periodic cases, on the whole real axis or in the whole space and in a finite interval - for all forms of fractional integro-differentiation. As a rule X is Lp -space or Holder space H>.. or a similar weighted space. As a matter of fact the representability of a function l(z) by the fractional integral I = ] 01 cp of order a studied in the book is a more important fact than the existence of fractional derivative of order a of a function l(z). We reveal general situations when the existence, in one or other sense, of a fractional derivative D01 I of a function l(z) is equivalent to the representability of the latter by a fractional integral. Then it is easy, in particular, to answer the question: why does the
xxiv
INTRODUCTION
existence of one or other form D01 f of a fractional derivative lead to the existence of a derivative D01 f, p < a, of the same form? Finally, one more distinctive feature of the book is our endeavour to unify the notation of various forms of fractional integra-differentiation. It is impossible to do without such a unification in a book containing many versions of fractional integrals and derivatives. The reader should immediately note the sign ± in the notation of fractional integrals and derivatives of functions of one variable. These signs mean the choice of the left-sided and right-sided fractional integra-differentiation connected with the left-sided and right-sided translations /( z =F t) , respectively. A consideration of both of these two forms is caused not only by the desire to achieve a common form of presentation, but mainly by the existence of interesting connections between the above forms of fractional integra-differentiation and the applications discussed in the book.
NOTATION OF THE M AIN FORMS OF FRACTIONAL INTEGR ALS AND D ERIVATIVES
If. cp I� cp I:+ cp Ib_ cp 1:+ ;9 cp , I:+;�" cp I:+ ;u ,, cp lb-;u,, cp IJ".a cp . K;,a cp I, , acp , K,,acp JOt cp J; ' >cp a-% cp J:+ cp I� cp I� cp , 1± , 8 cp aa cp Gi cp
1J± f
v:+J, v:_ J
5.2) 5.3)
- Liouville left-sided fractional integral (§ - Liouville right-sided fractional integral (§ - Riemann-Liouville left-sided fractional integral (§ - Rie mann-Liouville right-sided fractional integral (§ - fractional integrals of one function by another (§§
2.17) 2.18) 18.24, 18.38-18.41) - Erdelyi-Kober-type left-sided operator (§§ 18.1, 18. 2 ) - Erdelyi-Kober-type right-sided operator (§§ 18. 3 , 18.4 ) - Kober operators (§§ 18. 5 , 18. 6) - Erdelyi-Kober operators (§ 18. 8 ) - Riesz potential (§§ 12.1, 25.1) - Weyl fractional integrals of periodic functions (§§ 19.5 , 19.7) - Hadamard fractional integration (§§18.42-18. 44) - Griinwald-Letnikov fractional integral (§ 20.46) - Chen fractional integral (§ 18. 80) - Riemann-Liouville fractional integrals in the complex plane (§§ 22. 8 , 22.17-22.20) Bessel fractional integration (§§ 18. 6 1, 27. 8 ) - modifications of Bessel fractional integration (§ 18. 6 3) - Liouville fractional derivatives (§§ 5.6, 5. 7) - Riemann-Liouville fractional derivatives (§§ 2. 2 2, 2. 2 3, 2.32, 2.33)
xxvi
MAIN FORMS OF FRACTIONAL INTEGRALS AND DERIVATIVES
VC:+ ;u f D± f Do:/ DC:+ / , D6_ / 'D± f
v�a ) 1
D �a) I
-
fractional derivatives of one function by another (§ 18.29) Marchand fractional derivatives (§§ 5.57, 5.58, 5-.80) Riesz fractional derivative (§ 25.59) analogues of Marchand fractional derivative in an interval (§§ 13.2, 13.5) - Hadamard fractional derivatives (§§ 18.56, 18.57) - Weyl fractional derivatives of periodic functions (§ 19.17)
- Weyl-Marchaud fractional derivatives of periodic functions (§ 19.18) ) - Griinwald-Letnikov fractional derivatives (§ 20.7) !± - Riemann-Liouville fractional derivatives in the complex 'D';0 / , V± ,g / plane (§§ 22.3, 22.21) Chen fractional derivatives (§ 18.87) VC:f (E ± V ) a , (E ± D) a - modifications of Bessel fractional differentiation (§§ 18.71 , 18.72)
BRIEF HISTORIC AL OUTLINE
d��.P d).
-
f(t) cos(>.x - t>. + p1l' /2)
-
dt
(1)
in order to define the derivative for non-integer order. This was the first definition for the derivative of arbitrary positive order suitable for any sufficiently "good" function, not necessarily a power function. The examples mentioned above may be regarded as a prehistory of fractional integra differentiation. The proper history of fractional calculus began with the papers by Abel and Liouville. In the papers by Abel [1] (1823), [2] (1826) the integral equation
z
J
rp(t)dt --!(""),
( x - t)IJ
..,
:c
>
a,
0
0, was considered 74 years later by Sato [1] {1935) although its solution was in fact already known to Holmgren (1] {1865-1866). This latter paper by Holmgren should be especially noted. He used result (5) as a definition of fractional integration and gave a detailed investigation together with applications to the solution of ordinary differential equations. The merit of Holmgren's work is in the fact that he was the first who gave up the "complementary" functions and consciously suggested that one consider fractional differentiation as an operation inverse to fractional integration. Some years later Letnikov [1]-(4] (1868-1874), who was not aware of Holmgren's paper, expounded the theory of fractional integro-difl'erentiation from the same point of view. Holmgren's paper remained little known both to contemporaries and to later generations of mathematicians and therefore it was undeservedly little cited. In spite of which Holmgren was the first, after some rather formal arguments of Liouville, who gave a rigorous proof of Leibniz 's rule for the fractional derivative no ( uv) of a product of two functions. He also gave such a formula with a remainder in the integral form. Further, he was also the first to introduce the notion of fractional integration of one funct�on by another and gave a detailed investigation of the compositions of the form
(6)
where /; ( x) denotes an operator of multiplication by a function /; (x). Moreover he considered firstly partial and mixed derivatives of functions of two variables. Holmgren [2] {1867) also pushed well ahead in the application of fractional integrals to ordinary differential equations which was begun by Liouville (3] {1832). Griinwald [1] {1867) and Letnikov [1] {1868) developed an approach to fractional differentiation based on the definition
no f(x) =
( h /)(x) lim A . h0 h -0
(7)
While the arguments of the first author were rather formal, the latter gave a rigorous and thorough construction of the theory of fractional integro-differentiation on the basis of such a definition. Letnikov had in particular shown that thus defined n-o I coincides with Liouville's expression (4) for Rea > 0 and with Riemann 's definition under the appropriate interpretation of the fractional difference ( Ah f)(x). He proved the semigroup property within the framework of definition (7). A paper by Letnikov [2] (1868) was the first that contained a comprehensive historical survey of the development of fractional calculus. In a long paper (4] (1874) by Letnikov a complete theory of fractional integro-difl'erentiation was constructed on the basis of definitions {4) and (5). The detailed and comprehensive application of this theory to the solution of differential equations was also given. The reader may find in §42 the solutions of some ordinary differential equations which go badt to this paper by Letnikov. We note that there was an interesting pqblication by Sludskii [1] (1889) on the biography of Letnikov and his works. See also Nekrasov and Pokrovskii [1] (1889}. Many papers were published other than those of Liouville, Riemann, Holmgren, Griinwald and Letnikov. We may mention, for example, the papers by Peacock (1] (1833), Greatheed [1], [2] {1839), Kelland [1]-[3] {1840-1851), Center [1]-(4] {1848-1849), Tardy [1] {1858) and M.E. Vashchenko-Zakharchenko [1] (1861). Some of them contained polemics with predecessors. These were connected either with the idea of complementary functions or with the seeming contradiction between Liouville's and Riemann's definitions, though this contradiction seems
XXX
BRIEF HISTORICAL OUTLINE
far-fetched from the present-day point of view. Others developed or specified small points of the subject and do not contain fundAmental ideas. A further period in the history of fractional calculus is connected with the Cauchy fonnula
j(P) (z) = L 21ri
f (t -f(t)dt
{8)
z)P+1
c
for analytic functions in a complex plane. The direct extension of this fonnula to non-integer values of p leads to difficulties arising from the multivaluedness of the function (t - z)-P-1 and therefore it depends on the location of the curve £, surrounding the point z and on a cut C defining a branch of the function (t - z)-P - 1 . Such an extension was first made by Sonine [1] {1870), [2] (1872), who showed in the case of analytic functions that this new approach coincided ( with that of Riemann) for Rep < 0 (cf. (5)):
1 j{P ) (z) = __ ( -p) r
fz zo
f (t)dt
(z - t)1+P
{9)
'
the integration path being along the interval [zo, z] in the complex plane, where zo is the intersection of the curve £, with the brunch cut C. It should be emphasized that fractional calculus had developed from its origin in the complex plane, see for example the papers by Liouville (t]-[8] {1832-1837). The fonnula
(z (1:0 f)(z) =
�(:}
)a
1
J (1 - t)a j[(t - t)zo + tz]dt 0
in the complex plane, evidently being a modification of (9), had appeared in the paper by Holmgren [1, p.1] {1865-1866). Letnikov (3] {1872) made the important remark that £, being a circle, the Cauchy-Sonine formula (8) transformed into the fonn
+ J211' e-ipB f(z + rei8 )d0,
{l p) j (P ) (z) = r 21rrP
r = l z - zo l ·
(to)
0
We note that Sonine (2] (1872) continued the investigation of the Leibniz fonnula for Da (uv) which was begun by Liouville and Holmgren. We emphasize the priority of Sonine in the extension of the Cauchy fonnula (8) to noninteger p. Besides the development of the fractional calculus itself Sonine [4], (5] (1884) began the investigation of entities more general than fractional integrals, together with their applications to integral equations with special functions in the kernel. He obtained a solution of the Abel type equation z
J k(x - t)�P(t)dt = f(x), a
x>
a,
with an arbitrary kernel k(x) sat i sfying certain assumption (4], (5] {1884). In particular he found the solution of such an integral equation with the Bessel function in the kernel, the latter
xxxi
BRIEF HISTORICAL OUTLINE
frequently occurring in applications. We note that many years later this result was rediscovered by other investigators. Details concerning Sonine's ideas may be seen in Chapter 1, §4.2 (note 2.4), chapter 7, §39.1 (notes refering to §37.1 and §37.2) and Chapter 7, §39.2 (note refering to ·§37.3) below. In 1888-1891 Nekrasov [1]-(4) gave applications of fractional integro-differentiation in the form (8) to the integration of high order differential equations. He was also the first to give a procedure for reducing some multidimensional integrals to double integrals via the evaluation of compositions of the form {6). The latter represents the solution of differential equations considered by him. On the eve of the 20th century a comprehensive paper by Hadamaro [1] (1892) appeared. The idea of fractional differentiation of an analytic function via differentiation of its Taylor series
(k) Ck = J k(zo) ! •
(11)
although known before Hadamaro's paper, was used here as an effective working mathematical tool, understood as the fractional differentiation of an analytic function in a disk by a radial variable. Since then any method using (11 ) is usually named after Hadamard. We note that Hadamard dealt with fractional integration in the form of
JQ f (x) =
1
r�:) f (1 - e) Q- 1 f(ze)de,
(12)
0
which led him further to consider generalized fractional integrals of the form
1
f v(e)J(ze)de.
{13)
0
v
However Hadamard did not develop this idea, although he had considered the case (e ) = C - Ine)a- 1 . Many years later a substantial theory of generalized integration {13) was created by Dzherbashyan [4) {1967), [5) {1968). In 191 5 Hardy and Riesz [1] used fractional integration for the summation of divergent series. "Normal means" by Riesz, well known since then in mathematical analysis, represent fractional integrals of a partial sum of a series (see §14.8). The development of mathematical analysis and of function theory led to the appearance of new forms of fractional integro-differentiation. Weyl [1] {1917) defined fractional integration suitable for periodic functions:
rfaJ
I�a) rp ""'
00
L
k: - oo
(±ik)-a rpk eik �:'
IPO = 0,
(14)
It is realized as a convolution
I�a) rp = ..!_ 211'
2 11"
f w±(x- t)rp(t)dt 0
(15)
BRIEF HISTORICAL OUTLINE
xxxii with a certain special flUlction be written in the form
1 I+a rp = r("') ....
z:
\lf�(x).
He had shown that the fractional integrals
f (xrp(t)dt t)l-a - oo
__;_,�- ,
Q'
l_
-
00
f (t -rp(t)dt r(a) x)l-a '
rp = 1
(14)-(15) may
0. and in L p -spaces and in similar weighted spaces Ht (p) and Lp (p). Various notions and statements known in mathematical analysis which are repeatedly used throughout this book are discussed in the first section of the chapter.
§ 1 . Preliminaries We exhibit here certain ideas and propositions in mathematical analysis necessary for our purposes. Topics such as Holder weighted spaces Ht(p), the weighted spaces Lp (p). of summable functions, special functions and integral transforms are discussed. 1.1.
The spaces H >. and H>. = H>.. (n) the space of all functions which in general are complex valued, and satisfying the Holder condition of a fixed order A on n. It is simple to see that under such a definition only the case 0 < A � 1 is of interest, since if A > 1 then the space H>. contains only constant functions f ( x) = const . It follows from ( 1 .1) that /' (x) = 0 if A > 1 . In this connection (see Definition 1 .6 below) we will set H 0 (0) = C(O). Definition 1.1.
We shall also need the space
(1.2) of functions
f(x) satisfying a condition stronger than (1.1), namely f(x 2 ) - /(x i ) lx2 - x 1 l
--4
0
(1 .3)
for all X l E n. It is clear that h ). c H). . The space H1(0) is often called a Lipschitz space. We further give the definition of the space AC(O) of absolutely continuous functions. This space is wider than H 1 (0). Definition 1.2. A function f( x) is called absolutely continuous on an interval n, if for any e > 0 there exists a 6 > 0 such that for any finiten set of pairwise nonintersecting intervals [ak , bk ] c n, k = 1 , 2, . . . , n, such that E (bk - ak ) < 6, k= l n the inequality E. l f(bk ) - /(a k ) l < e holds. The space of these functions is denoted k= l by AC(n) .
3
§ 1. PRELIMINARIES
It is known (see the books of Kolmogorov and Fomin [1, p.338] or Nikol'skii [8, p.368-369]) that the space AC(O) coincides with the space of primitives of Lebesgue summable functions:
f(z) E AC(O)
:1:
f(z) = c+ J cp(t)dt,
b
J lcp(t)l dt
.( Ot ) and f(z) E H>.(02 ) and f(c - 0 ) = f(c + 0) , then f(z) E H>.(O). One may find a proof of Lemma 1 in the book of Muskhelishvili [1 , p.21] , for
L emma
example. We shall also need the following Holder weighted spaces.
Let p(z) be a nonnegative function. The space of functions f(z) such that p(z)f(z) E H>.(O) is denoted by H>.(p) = H>.(O; p). In what follows a weight function p(z) will be a power function coupled to a
D efinit io n
1.4.
finite number of points
p(z) = IT l z - zk l11 k , n
where P.k are real numbers and
ZA:
E
k =l
0.
( 1.7)
The case
( 1 .8) will be the most prevalent. If 0 contains the point at infinity then it is expedient to take the weight ( 1 . 7) in the form n
p(z) = ( 1 + z2 )11/ 2 IT lz - Xk jll k , k =l
i.e. to couple it to the point use the notation
x = oo.
( 1 .9)
While considering the weight ( 1 .9 ) we shall
( 1 . 10) the exponent of the weight at infinity. By the definition of the space functions in this space are represented by the form
t(x) =
;. (p), ( 1 . 11 )
H>.(p).
1.5. Let p(z) be given by ( 1 .7) or ( 1 .9 ) . We denote by H6 (P) = Ht(O; p) a set of those functions in H>.(p), for which fo (zk ) = 0 and /o(oo) = 0,
D efinit io n
§ 1 . PRELIMINARIES
5
the latter in the case when 0 contains the point at infinity, in the representation ( 1 . 1 1) . We denote by Ht a space of functions from H >. , which vanish at x = a and X = b. ·
We note that we shall often write HA , H>. (p), H6 (P) instead of H>.(O), H>.(p, 0), H6(p, 0) in those cases when there is no chance of misunderstanding. We shall also use the weighted spaces h � (p)
=
{ f(x) : p( x)f( x ) E h>. , p(z) / (z)l �=a
=
p(x ) f ( x ) l �= b
=
0},
( 1 . 12)
where h>. is the space (1.2) , (1.3) and p(z) is the weight (1.8). The spaces we have introduced are linear spaces easily equipped with norms. Thus, when 0 is an interval we put (1. 13)
The second term in (1.13) is an infimum of all possible values of a constant A in ( 1. 1) . The space H>. is complete with respect to the norm (1 .13), i.e. it is a Banach space, the proof being given for example in the book of Muskhelishvili [1 , p.173] . When 0 is a whole line of a half-line the norm is introduced on the bases of ( 1 .6) by the relation
It is possible to make sure of the completeness of H>. (O) with respect to the norm (1 .14) , for example, by mapping the whole line (the half-line) onto the circle (half-circle) by means of a fractional-linear transformation and using the completeness of H>--space for any bounded curve - Muskhelishvili [1, p.173] . The space ( 1.2) is also known to be a closed subspace in H>. - Krein, Petunin and Semenov [1, p.269]). In the weighted case a norm is introduced on the basis of ( 1 . 1 1) by ( 1 . 15) the completeness of H>. (p) with respect to this norm being obvious owing to the isometry ( 1 . 15) between the spaces H >. (p) and H >. .
6
CHAPTER 1 . FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
The following useful property holds: if f(x) E HA([a, b]) and 0 < a < A, then
- f(c) E -a g(x) = f(x) lx - cla HA ([a, b]), a � c � b ; llu iiH�-o
( 1 . 16)
� K II / IIH� ,
where K does not depend on f(x), see, for example, Muskhelishvili [1, p.22] . Let L I (n) be a space of Lebesgue integrable functions on n. If n is a finite interval and p(x) is the weight (1 .7) the following imbeddings ( 1 . 17) are valid together with the norm inequalities ( 1 . 18) provided that A � J.L k < A + 1, k = 1, 2, . . . , n (see the definition of 1 1 / ll £1 in (1 .26) ) . Here and below cm (n) denotes a space of functions which are m times continuously differentiable on n with norm
m
L: /( ") (z ) l , 11 / llcm = max xe n O 1
m = 0, 1, 2,
.
.
.
,
( 1 . 19)
k=
I IIII co
=
lillie ·
We denote by C[f = C[f (R1 ) a space of infinitely differentiable finite functions in R1 . The space introduced below is the extension of the space HA(Q) to the values A > 1. Definition 1.6. Let A = m + o- , f(x) E HA(f!), if f(x) E Cm (n)
where m = 0, 1 , 2 , . . . and 0 < and f(m) (x) E H u (n) ; and
o-
< 1.
We say that (1 .20)
When A is an integer, one has often to deal with a somewhat wider space of functions with the Holder (Lipschitz) condition containing a logarithmic multiplier as in Theorem 3.1 and 3.2 in § 3. In this connection we give the following definition.
7
§ 1. PRELIM INARIES
Let A = m + u, where m = 0, 1 , 2, . . . , and 0 < u < that f(x) E H A ,k = H A ,k (O) , k E Ri , if f(x) E cm (O) and
Definition 1. 7.
1
lh l < 2 '
1.
We say
( 1 .21)
and 1 1 / I IR;\,,.
lf(m)(x + h) - f(m)(x) l sup = 1 1 / llcm + x,x+hEO k lhl�l /2
( Jh)
l h lu ln
By analogy with ( 1 .2) we introduce the space h A ,k which satisfy a condition stronger than ( 1 .21), namely
= hA ,k (0)
( 1.22)
of functions
( 1 .23)
Similarly we have the weighted spaces of functions H;• k (p) , H A•k (p) and h �· k (p) analogous to ( 1 .5 ) and ( 1. 1 1) and (1.12 ) . Remark 1.1. Although it is very convenient to use Holder spaces in various problems, (and they are widely applied in this book) , they have one essential shortcoming: they are not separable. There exist no "good" dense subsets in H A and If A (p) and it is impossible to approximate the function /( x) E H A by "better" functions in the norm of the space H A , since a closure of "good" functions in the norm H A gives h\ but not H A , (see, for example, Krein, Petunin and Semenov [1, p .269]). This "negative" property of the space HA will nowhere be displayed in our considerations. However we should bear it in mind if we try to construct approximation methods for the solution of equations studied in Chapters 6 and 7 in those cases when they are considered in H A or H>.(p). The integral Holder spaces n; will be considered in § 14. The proximity of the values of f(x 1 ) and / (x2) will be estimated not in a uniform metric as in ( 1.1 ) , but in an integral one. 1 . 2.
The spaces Lp and Lp (p)
We assume that the reader is familiar with Lebesgue measurability of functions and the Lebesgue integral. Again let n = (a, b], -oo :5 a < b :5 oo. We denote by Lp = Lp(O) the set of all Lebesgue measurable functions / (x) , complex valued in general for which J 1 / (x) IPdz < oo, where 1 :5 p < oo. We set 0
8
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
= { [ 1/(z) l' dz } ( 1. 24) If p = oo the space L,(O) is defined as the set of all measurab le functions with a finite norm ( 1.25 ) 1 1/IIL ..., (O) = esssup 1/(z) l , zen where esssup 1/(z) l is an essential maximum of the function 1/(z) l see details in Nikol'skii [6, p.12-13 ). Everywhere be low we assume that 1 � p � oo. As usual two equivalent functions, i.e . differing on a set of zero measure, are considered to be equal to one element of the space L, (O) . That is, they are not distinguished as e lements of this space. For norms ( 1.24) and ( 1.25) we shall also use the notations ( 1.26 ) 1 1/11, = 11/IIL,. = 1 1/IIL ,.(n) · Let us give some properties of the spaces L, : a) The Minkowsky inequality ( 1.27) II/ + Y IIL,.(O) � 1 1/IIL,.(O) + I IYIIL,.(O) , so that L, (O) is a normed space. It is also known that L,(O) is a complete space; b) The Holder inequality l/p
1 1/IIL.(n)
-
j n
1/ ( z )g (z) l dz � 1 1 /IIL,.(n> IIYIIL ,. , (n) ,
p' = pf(p - 1 ) ,
(
1.28)
where (z) E L,(O), g(z) E £,, (0). Index p', which is connected with p by the relation/ 1 1 ( 1.29) p- + p'- = 1, is called conjugate to p. We note that ( 1.28) is true if 1 � p � oo (p' = oo, if p = 1, and p' = 1, ifp = oo) . From ( 1.28) the generalized Holder inequality J lit (x)
n
·
· · fm (x )ldx �
li lt I IL,.1 (0) · · · 11 1m IIL,.,. (n) ,
(
1.30)
§ 1 . PRELIMINARIES
follows where b: (x) E
9
Lp,.(O), k = 1, 2, . . . , m, kE= l 1/pk = 1. m
We see that the imbedding
(1.31) n being a finite interval; is derived from Holder inequality c) Fubini 's theorem which allows us to interchange the order of integration in repeated integrals:
(1.28),
Let !1 1 = [a , b], !1 2 = [c, d] , -oo � a < b � oo, _;oo � c < d � oo , and let f(x, y) be a measurable function defined �n !1 1 X !1 2 . If at least one of the integrals
Theorem 1.1.
J dx J f(x, y)dy, J dy J f(x, y)dx , JJ f(x, y)dxdy,
n1
n�
n�
n1
n 1 x n�
is absolutely convergent then they coincide.
The following particular case of Fubini's theorem holds, namely b
b
z
b
J dx J f(x , y)dy = J dy J f(x, y)dx a
a
a
(1.32)
y
assuming that one of these integrals is absolutely convergent. The latter relation is called the Dirichlet formula.
The generalized Minkowsky inequality
adjoining the Fubini theorem is also true; d) the property of mean continuity for functions in Lemma 1.2.
Let f(x) E Lp(O),
1�
p
0, (1.41) holds. In particular, setting � = t -1 , we come to the conclusion that any homogeneous function of degree a may be represented in the form (1.42) Theorem
1.5.
Let k(z, t) be a homogeneous function of degree -1. If 00
00
K = j l k (z, l)l z - 1/p ' dz = j l k (l , t)1 t - 1 1P dt < oo, 0
(1.43)
0
then the integral operator K
• 1 f formula by the c�c = t1 B: (a), see Luke (1, p .20]. Generalized Bernoulli polynomials may be found in Gel'fond [1, Chapter 4]). In some cases we shall also use the ol garithmic derivative of the gamma function, called by the Eu ler psi-function co = 1,
I
oo ,
d r(z) = r'(z) . ,P(z) = dz r(z)
In D. The beta-function B(z, w) . The Eu ler integral of the first kind 1
B(z , w) = J �z -1 (1 - z)w -1 dz,
Rez > 0, Rew > 0,
(1.67)
(1.68)
0
is
called the beta-function. It is connected with the gamma-function by the relation
B(z, w) = r(z)r(w) (1.69) r(z + w) " The integral (1.68) makes sense when Rez = 0 or Rew = 0 (z #:- 0, w #:- 0). In this case it is understood to be conditionally convergent. In particular, there exists the limit 1- t B( z, iO) = t-O lim j zz - 1 (1 - z)18-1 dz, Rez > 0, f:. 0, (1.70) which coincides with the analytic continuation of B(z, w) with respect to onto the values Re w 0, w #:- 0. (J
0
=
w
18
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
The integral
00
J (t - z)a- 1 (t - y){J- 1 dt
=
(z - y) a+{J- 1 B(a, 1 -
Cl' -
{J),
(1.71)
z
z > y, 0 < Rea < 1 - Re{J, which is reduced to later.
(1.68)
by the substitution
t = y + (z - y)e - 1
will be useful
The Gauss hypergeometric function is defined in the unit disk
E. of the hypergeometric series
� (a)t( b) t zk 2 F1 (a, b; c; z) = L.J (C) k k' . k=O
as
the sum
(1.72)
•
Its parameters a, b and c and the variable z may be complex (c :/; 0, -1, -2, . . . ) and (a) t is the Pochhammer symbol (1.45). The series is convergent for l z l < 1 and for l z l = 1, Re(c - a - b ) > 0. For other values of z the Gauss hypergeometric function is defined as an analytic continuation of the series. One of the methods of such a continuation is
the Euler integral representation
1 6- 1
1 . 4.
As
§ 1. PRELIMINARIES
23
Integral transforms
is known classical integral transforms are of the form 00
(K 0, is defined follows as
a
00
c,o* (s) = rot{c,o(t);s} = J t•-1 c,o(t)dt, 0
(1.112)
and its inverse is given by the formula
-y+i oo
c,o(x) = rot-1 {c,o* (s);x} = 2�i J c,o* (s)x - •ds, = Res. -y-i oo "Y
(1.113)
26
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
These relations are derived from (1.102) and (1 . 104) if we replace Po ·
(1.120)
By using the Mellin transform (1.112) one may obtain another form for the inverse
28
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Laplace transform
"Y +ioo
{L - 1 g)(z) = 2�i j ")' ioo -
f{�: s) g*(l - s)ds,
Res = 'Y < 1.
(1.121)
see Marichev [10, formula (8.29)]. The integral
X
[h and suppose that equation considered on a finite interval [a, b]. The factor 1/f(o:) is chosen for convenience by reasons which will become clear below. Equation (2.1) may be solved in the following way. Changing to t and t t) - a and to s respectively in (2.1), multiplying both sides of the equation by ( X
x -
0
i$
oo
-
x
x -
integrating we have
I X
a
dt
(x -
f(t)dt (s)ds t)a I (tft?- sp -a = (a) I ( t)a r
t
X
a
a
x -
(2.2)
·
Interchanging the order of integration in the left-hand side by Dirichlet formula (1.32) we arrive at
(a:) I ( f(t)dt I ft?(s)d I ( t)a The inner integral is easily evaluated after the · change of variable t = s + T ( - s) and application of the formulae (1. 6 8), (1.69): 1 ( -a a-1 = I Ta - 1 (1 - T) -a dT t) (t s) dt I = B{o:, 1 - a:) = f(o:)f(1 - ) X
X
8
a
a
x -
dt t)a( l - sp -a =
r
X
a
x -
·
x
X
x -
0
$
a: .
Therefore
f(t)dt . = ( s)ds ft? r(l - a:) I ( t)a I X
a
1
X
a
x -
(2.3)
30
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Hence after differentiation we have: :t:
1 is reduced to the case < a < 1, by differentiating both parts in (2.1). The solution of Abel equation with a > 1 is in fact contained in Theorem 2.4. Quite analogously the Abel equation of the form
0
0
b
1 1, and that the operator V�'+ is bounded in the space L, , p > 1 . In Theorem 2 . 2 below we give sufficient conditions for the existence of fractional derivatives of an arbitrary complex order a , Rea � 0, the simpler cases 0 < a < 1 and Re a = 0 being considered apart in Lemmas 2.2 and 2.3 below. Since the theorem will be stated in terms of the class AC" (see Definition 1 .3 ) , we give first the characterization of this class.
/(z)
space b]) consists of those and only those functions /(z), which are Therepresented in the form AC"([a,
Lemma 2.4.
( n _ 1 )!
where cp(t) E
��(z - t)"-1 cp(t)dt
n- 1 /(z) = 2: c�:(z L 1 , b) , c�: being arbitrary constants. 1
a
+
a)A: .
( 2.41 )
A:=O
(a
The proof of lemma follows immediately from the definition of the space AC" ((a, b]) and from ( 1 .4 ) and ( 2.16 ) . Note that in ( 2.41 ) we have
( 2.42 )
Let Re a � 0 and E AC" ((a, b]}, n [Rea] + 1 . Then v:+J e�ists almost everywhere and may be represented in the form
Theorem 2.2.
/(z)
=
��
n - l J( A: )(a) -a f(n )(t)dt 1 a ( z - a) A: + 'Da + f = r(n - a) ( z - t) a - n + l . A:-0 f(l + k - a) a
�
( 2.43)
40
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
Proof. Since
/ ( z) E Acn , we have the representation (2.41) . Substituting it into
(2.32) and taking (2.42) into account we obtain (2.43) after simple transformations. It is not difficult to see that (2.20), (2.21) remain valid for complex values of a, /3 if Re a > 0 , Re/3 > 0, (and 1/p + 1/q < 1 + Rea for (2.20)). The same concerns Theorem 2. 1 , Lemma 2.1 and the Corollary of the latter with 0 < Re a < 1. Note at last the validity of the following lemma.
Let 0, denote the space of functions l(z), �presented by the left-sided fractional integral of order a of a summable function: I = P;+ cp, cp E Lp(a, 6), 1 � p < oo. Definition 2.3.
The characterization of the space 1:+ ( L1) is given by the following theorem generalizing Theorem 2.1. Theorem 2.3.
sufficient that
In order that f( z) E I:+ (Lt), Re a > 0, it as necessary and
fn- a ( z) ";;JI';;:; a I E AC" ([a, 6]) ,
(2.55)
f�i:Ja (a) = O, k = 0, 1, 2, . . . , n - l .
(2.56)
where n = [Rea] + 1 and lhat
Proof. Necessity. Let f = 1:+ cp, cp E L1 (a, b). In view of the semigroup property (2.21) we have r;;:; al = 1::+ cp, cp E L1(a, b), and the conditions (2.55), (2.56) follow then from Lemma 2.4. Sufficiency. Conditions (2.55), (2.56) being satisfied we can represent fn- a ( z) , according to Lemma 2.4 as ln- a ( z ) = 1::+ cp, where cp E L1(a, b). Consequently I:+a I = 1::+ V' = I:;:; a 1:+ cp owing to the semigroup property (2.21). Hence I::; a (l - 1:+ cp) = 0. Since Re (n - a) > 0 we have I - I:+ V' = 0 by lemma 2.5
which completes the proof. • Let us emphasize in connection with definition 2.3 that the representability of a function l( z ) by a fractional integral of order a and the existence of a fractional derivative of this order for f(z) are different things. Thus the function
44
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
f(z) = (z - a) a-1 , 0 < Re a < 1, already familiar to us has a fractional derivative which is equal to zero - ( 2.35 ) . However the function ( z - a ) a- 1 may not be
represented by a fractional integral of order a for any summable functions. The reason for this is that It -a ( a ) =:fi 0, in the case of this function, so the conditions (2.56) are not satisfied. The reader acquainted with distribution theory will understand that the function ( z - a ) a-1 may be the fractional integral of order a only of a distribution, namely the Dirac delta-function 6(x - a ) - see § 8.1. Let us focus on the idea itself of the existence of a fractional derivative. For simplicity let 0 < Re a < 1 . If we say that VC:+ f = (d/dx)I;f. a f exists almost everywhere, then we must take into account the following. It is known that the existence of a summable derivative g' ( x) of a function g( x) does not yet guarantee �
J g'(t)dt =F g(x) + c in general - see a for example Natanson's book [ 1, p. 199] . Moreover, there exists ( ibid., p.201 ) such a monotone continuous function g(x) 't canst that g'(x) = 0 almost everywhere. the restoration of g(x) by the primitive, i.e.
This has already been mentioned in the proof of Theorem 2.1. These "exotic" effects are removed if we deal with absolutely continuous functions. We remind the reader that integration by parts in the Lebesgue integral is in general possible only for absolutely continuous functions, the latter being already employed in the proof of Theorem 2.3. For these reasons it is clear that the supposition " the fractional derivative VC:+ f exists almost everywhere and is summable" is insufficient for developing a satisfactory theory, i.e. it is insufficient for a function f(x) to be represented by a fractional integral of order a . So it is necessary to make this supposition stronger. For this purpose we give the following
Let Re a > 0. A function f(x) E L 1 (a , b) ·is said to have a summable fractional derivative VC:+ f, if 1:.; a f E ACn ([a , b]), n = [Re a] + 1 .
Definition 2.4.
I n other words this definition uses an idea employing only the first of two conditions ( 2.55 ) , ( 2.56) characterizing the space IC:+ (L 1 ).
Remark 2.2. If VC:+ f = (d/dx) n 1:.; a f exists in the usual sense, i.e. 1:.; a f is differentiable n times at every point, then, evidently, f(x) has a derivative in the
sense of Definition 2.4. We have found necessary to dwell in detail on the above reasons and to give, in particular, Definition 2.4 because confusing the two ideas - existence of the fractional derivative and representability of a function by the fractional integral and sometimes the careless interpretation of the first of these notions, has caused errors in the papers of many authors. The following theorem which is the main one in this subsection deals with the question mentioned in its heading. Theorem 2.4.
Let Re a > 0. Then the equality (2.57)
u
§ 2. RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND DERIVATIVES
45
valid for any summable function cp(z) while (2.58)
u
satisfied for (2.59)
If we assume Jthat instead of (2.59) a function /(z) E L1(a,b) has a summable ·;,trivative v:+ (in the sense of Definition e.4), then (2.58) is not true in general and is to be replaced by the result (z - a)a- t- 1 Ia+a va+a I - /(z) - � f:'o r( - k ) fn.. (x - t) 01- 1 dt a
(3.2 ).
is true. Changing the variable t = a + s(x - a) we deduce We prove that 1/J(x) E The cas� or .,P(x) E considered first. For brevity we set g(x) = cp(x) - cp(a),' so that
n>..+a
n>..+a, t .
A + a � 1 is to be
(3.3) Let
h > 0; x, x + h E [a, b) .
1.,P(x + h) _ .,P(x) = f(a)
We have
(J
x -a
-h
g(x - t)dt _ (t + h) l -a
J g(xt 1 --at) dt)
x -a 0
a -1= f(1g(x) + a) [(x - a + ht - (x - a) ] + f(a) 1
+ f(a)
0
J g(x(t-+t)hp--g(x)a dt
-h
. J [(t + ht - 1 - t01- 1 ][g(x - t) - g(x)]dt
x -a 0
(3.4)
§ 3. THE FRACTIONAL INTEGRALS OF H O LDER AND SUMMABLE FUNCTIONS
55
If h � z - a, then according to (3.3) we find
If 0 < h < z - a, then by the estimate
(3.3) and the inequality ( 1 + t) a - 1 5 o:t , t > 0, we have
Then
Finally we estimate
J3:
(3.5) Hence the estimate I J3 I 5 ch>. +a , � + o: 5 1, holds if z - a 5 h. If z - a > h and � + o: < 1 we have I J31 5 ch>. +a also, in view of the convergence of the integral (3.5) at infinity, since
if t
>
1. If � + a
= 1 then (3.5) yields the estimate
56
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
provided that 0 < h < 1/2. Collecting the estimates for J1 , J2 , Ja we complete the proof of the theorem in the case � + a 1 .
$
We find i t more convenient to give the proof of the remaining case � + a > 1 in another place after introducing and investigating the so-called Marchand fractional derivatives. So we refer the reader for the proof of the case � + a > 1 to § 13.4 see the text following Lemma 13. 1 . • The case of an arbitrary a > 0 and � > 0 will be considered in Theorem 3.2 below. As a preliminary we give some corollaries from Theorem 3.1.
-
Corollary 1.
The operator X
cp(t) - cp(a) r (a) j (x - tp- a dt, 0 < a < 1 , a
-1-
is bounded from H>., 0 $ � � + a = l.
$ 1,
into H>. +a if � + a :f 1 and into H >. + a, l if
The operator I�+ is bounded from C = H 0 into H0• It is easy t o see, indeed, that /�+ is bounded from L oo into n a :
Corollary 2.
r (a)lf(x + h) - f(x) l $ x +t
{ J [(x - tt- l - (x + h - t)0- 1 ]dt X
sup
a<x.+cr
H �+ a , l
if � + a is not an integer or if � and a are integers if � + a is an integer but � and a are not integers.
57
.
Theorem 3.2 is deduced from Theorem 3.1 if we take into account that the function
] m ,P(z) = r(la) J [.- 1-' dt < ( x + h - tp-a - ch"' ( x·+ h - t) l -a
J
J
a
a
> 0 and
z (t - a)>. + + < c(x - a)"' (x - t) l -�-'dt a . a -< ch>. a a if p
h we have
z (t - a)>. - 1-' dt ch ch < ch>. +a I J2 1 .+ a I J2 1 -< (h + x - a) l - 1-' ( x - tp-a - (h + x - ap- >. -a - ch a
J
ch
if JJ
> 1 according to (3.9) and (3.8) respectively. Finally, changing the variable t = a + s( x - a) we obtain
J 1
I(
I Ja l $ llgiiH• ( z - a)"'+ " 1 •.>. - p - •"' I 1 - s + 0
.,
: a ) a- ! - { 1 - •) "-' I ds.
60
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
If z - a
� h, then I J3 1 � c ii Y IIH � h>. +a and if z - a > h then
Collecting the estimates for Jt , J2 , J3 we have
Using this estimate and the inequality
G(z) E n;+ a .
I G1 (z) l � c llg ii H� (z - a)>.+a we deduce that
2. The case p(z) = (b - z)", v > .\ + a . Now . and g(a) = g(b) = 0 in accordance with Remark
(b - x) - " g(z) where
3.1. We are to prove
that
X
( )
G(z) = f bb -- zt a
II
g(t)dt +a (z - t) l -a E n>- ' X
G(a) = G(b) = 0. Since I G(z)l � c J(t - a)>-(z - t) a - l dt as x --+ a, a the condition G(a) = 0 becomes obvious. If z --+ b, then
and that
I G(x) l � (b - z)"
then
b-a
J t>- - "(t + z - bt- 1 dt.
b- x
Hence after the change of variable t = ( b - z)e we have
I G(z) l � (b - z) >.+a
,_ _ b-'i'
J1 t>.-v (t - l )a- l dt
� (b - X) >. + a f t " - >. (t dt- 1 ) 1 -a 1 00
and so
G(b) = 0 .
To prove the Holder property of the function
G(x) we represent
§ 3. THE FRACTIONAL INTEGRALS OF HO LDER AND SUMMABLE FUNCTIONS
61
it in the form :c
:c
f ((bb - x)" - (b -l-t)" g(t)dt G(x) = f (xg-(t)dt + - t)" (x - t) a tp-a a
Gt (x)
a
I Gt l n A+ar � ciiYIInA by Theorem 3.1.
Here E H �+a and x + h E ( a, b) for ( z ) we have:
G2
Assuming that
where :c+h
Jl = f (b(b -- t)X "-(xh)"+ h- -(b1)-1t)"- a g (t)dt, :c
J2 = [(b - h -
z " -
)
:c
(b - z)"]
j (b
_
a
:c
(
� ��
t)" (
_
t) '- ,
J = f (b - x)"(b --t)(b - t)" [(x + h - t)a- l - (x - t)a- l]g(t)dt. 3
"
a
Using the estimate
lg(t) l � I YI nA (b - t)� and (3.8) we obtain :c+h
I Jt l � c J (x + h - t)a (b - t)�- 1 dt :c
h
= C f ea (b -
X-
h + e ) � - l c.Ie
0
from whence after the change of variable . e = ( b - X - h )s it folloWS that h/(b-:c-h)
I Jtl � c(b - X - h)�+a J 0
s a ( l + s) � - 1 ds.
62
CHAPTER
1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
If b - z - h 5 h , then
[
]
h/(IJ- z - h)
j
I Jt l � c(b - ., - h) �+ 1 +
.� +"' - 1 ds
1
� chH"' .
But if b - z - h � h , then
J
h/(6-z-h)
I J1 I 5 c(b - z - h)>. +a
s a ds � ch>. +a .
0
For J2 , applying (3.8) again we have
J z
I J2 I � ch(b - z) "- 1
(b - t)>. -v dt (z + h _ t p -a
a
1 _ < - c h(b z ) "-
J( z
(b - t)>.-v dt . z - t ) 1 -a
a
00
Now we change the variable z - t = (b- z)e. Then I J2 1 � ch(b- z) >-+a - 1 I ea- 1 (1 + 0 e>>.-v cJe since II > � + a. Hence, IJ2 1 � ch>- +a . It remains to estimate J3. By substituting t = b - s(b - z) we have
T l l·�-� 1 ·- ·
I Jal � c
(b - o: ) �+ "'
1
l ( l + �., r-· 8 -
b
l
(s - 1)"' - ' ds llgi i H • .
Applying (3.9) we obtain
h
I J3I � cII g II H" ( b - z) >.+a b - z
0-'i' ·-·
J1
1 1 - s" l ds � ch >-+a s "->-(s - 1)2-a
since b - z � h and � + a < 1 . The theorem is thus proved. • b A similar statement for the weight p( z) = ( z - a ) ( - z )" is easily derived from Theorem 3.3. Namely, the following theorem holds.
P
§ 3.
THE FRACTIONAL INTEGRALS OF HOLDER AND SUMMABLE FUNCTIONS
63
Let 0 < � < 1 and � + a < 1 . The operator 1:+ is bounded from H6(P) into n; + a (p) with p(x) = (x - a)11(b - x)", p. < � + 1, v > � + a.
Theorem 3.3' .
Proof. Let functions
2 :5 ch01+ P if z ;:::: h. If z :5 h , then
4> 2 :5 2z"Y +P
I 1
t - "Y (t + z)01- 1 dt
0
= 2z01+P
I s-"Y(s 1 /z
+ 1)01 - 1 ds :5 ch01+P .
0
Collecting the estimates for ct> 1 and . (x - t ) l -ex f= 0
((b - x)>.+ex )
3.3. Mapping properties in the space LP
Fractional integrals are known to keep, at least, the space Lp (a, b) invariant, see (2.72) and (2.73). The statements revealing how much the fractional integral I:+
Let 0 < a < 1 , r.p Then the equalities
E
Lp (O, 1),
0
0, � ;:::: 0, b > 0 hold. The former is proved by simple estimates after
changing the order of integration on the left-hand side. The latter is derived from the former if we replace z by z - 1 , b by b - 1 and ap - 1 and llk > (a - ; - m) if J.lk ap - 1, k 2, . . . , n. Then the operator I:+ is bounded from Lp (p) into Lq ( r) . We investigate the problem whether the fractional integrals I:+
Q -
1/p, then, after the change of variable
l v(x) l $ (d - zt - t ii'PoiiL,
In the case when
(f
"'·--·:tel Ot
t = z - e(d - z), we get
e(a- l )p' ( 1 + e) - vp' de
)
1/p'
v = a - 1/p we similarly obtain 1
i v(x) i :«; (d - xJ " II'PoiiL , (J e( a- l )p' de + :5
0
c(d - z)" ll<poiiLp (l + l ln(d - z) l ).
j C ' de)
� tl-:te
1
since we take z - a 1 > d - z. It follows from the above estimates that f(d - 0) = 0. It is similarly proved that /( d + 0) = 0 if d < b. To estimate the Holder norm of the function /(�) when z E [ab dj, we
80
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL
represent f(z) as z)" - (d - t)" cpo (t)dt + j (d/(z) = J (zcp-o(t)dt a t)"(x - tp - a tp (da a = ft (x ) + h (x). By Theorem 3.8 we have ll!t llnar-1/P ((a,b]) $ cllcpo i iLp(a,b) · For /2 we get !2 (x+h) -!2 (x) J1 + J2 + Ja, $ x < x+h :::; d, J�, J2 and Ja being the same as (3.7) but with (z - ) replaced by (d- x)". We estimate J1 considering for simplicity the case v :::; 1 only. We have x+h 1 ( ' p ) ' l ( ) llt l � II 'Po i iL, J ( d- W"• (z + h - t) a - l ( d - t)" - (d - z- h ) " ( X
in
X
=
a1
a �'
dt
X
x+h ( � cllcpo i i L, J ( d - W"P ' (z + h - t) (a+ v - l )p'
dt
X
)
P'
� •
x+h 1 ( ) p ( ' � cii'Po i i L, J (z + h - t) a- t dt) � ch"- } II'Po iiL, · P'
X
Estimating J2 we find
a1 1 p t)-v t)-"P dt dt ) pr (! (d(d$ ch" llcpo iiLp (al t)(l - a)pl + J (x t)(l - a)p1 a a1 I
X
_
I
_
The integral in parenthesis coincides with the integral in (3.27) which has already been estimated. Therefore, we have IJ2 I :::; ch" llcpo iiLp(a ,b) if v < a - 1/p. Ifl vp � a-1/p, then the expression in square brackets is estimated by 1+ c(d-x)a- v - f :::; 1 + cha - v - l /p if v > a - 1/p and by 1 + I,. e r( c:r 1 l{p) , see Mihailov [1, p.29). Let us also note that in Davis [1 , p.105) (see also the equation
0. We introduce the notation
- absolutely monotone functions. The definition of this idea is based on the fractional integra-differentiation of the form Further generalizations of absolutely monotone functions were given in this way by Saakyan [1], [2] and Dzherbashyan and Saakyan [2]. We note also that in the latter papers a generalization of the Taylor expansion was suggested, which was associated with the Mittag-Leffier function and with generalized fractional
(4.10).
n -1 n ( V�� + Aj E)!, j =O
differentiation of the type 0
< p < 1 by Saakyan [3].
p
>
1,
which was ext�nded later on to the case
(3] dealt with the Taylor expansion of the form f(z) = + + E (l>a d f)(zo)(z - zo)a d /r(l + + ak) in the complex plane, the particular case of k = - oo such an expansion being earlier considered by Fabian (3]. We refer also to Osler (6] where a We remark also that Osler
00
Ot
certain integral analogue of the Taylor expansion in the complex plane is given. In § 7.3 below we discuss a certain such integral analogue, different from that considered by Osler (see Remark 7.3) . In connection with the generalization of the Taylor expansion (2.63) , Badalyan (1] obtained a relation of the type (4 . 11) for constructions more complicated than fractional derivatives. 2 .9. The idea of differentiation with respect to the power exponent widely employed by VolteiTa [2] for evaluating integrals of power-logarithmic functions was developed by Rubin [10], [14]. One may follow Rubin (10] and apply fractional integration with respect to {3 in order to evaluate fractional integrals of the functions (x - a)P-1 1n11(1/(x - a)) (for x - a < 1). The paper by Rubin (14] contains the generalization of this approach based on the application of the convolution operation with respect to the variable {3. One may obtain in this way relations for evaluating fractional integrals of functions of the type
Ink t = !nln.... . . h) t, -oo < Ak < oo, {3 > 0. k
(x - a)P-1
TI {Ink x.:a ) >.., , where
k=O
2.10. Love (4] gave sufficient conditions for the existence of the fractional integral (2.38) of purely imaginary order. He showed, for example, that if a function f is integrable over
J t -1 wt (!, t)dt < oo, 6
[0, oo] and the condition
S
>
0,
is satisfied, where
w1 (!, t)
is the integral
modulus of continuity of f (see (13.24) below), then the fractional integral Po� exists for any real 9. Corq.pare this with Theorem 13.5, which contains sufficient conditions for the existence of the fractional derivative l>:+ J. It was also shown by Love [4 ; 5, p.388r that ��� � exists for a 0
J E Lt(a, b) if and only if I�+'' J E AO([a, b]); then Po� ! E Lt(a, b) and the relation I;!' JA� f = J holds almost everywhere on (a, b).
function
90
CHAPTER 1. FRACTIONAL INTEGRALS AND DERIVATIVES ON AN INTERVAL 2 .11. The estimate (2.72) is a particular case of the inequalities
0, 0
b),
�
0,
g(t) E cm([o,b]), m 1, g(k)( t) � where cx > 11 > and II l iP is the norm in Lp(O, k = 1 , . . . , m - 1, g(m)(t) � -a � 0 (a > in the second inequality). These inequalities were proved by Bukhgeim [1], [2, p.46] and used by him in (2] in the investigation of the inverse problems of reconstructing differential equations by given traces of their solutions. 1]) and let {la } a >O be a family of linear 2.12. Let X = Lp(O, 1), 1 � p oo, or X = operators in X. Is it true that the conditions
0
Onent is variable (see RemMk 3.5) is as follows. Let 1-' 1 p - 1, 0 1-'k p - 1, k = 2, 3, , n - 1, 1-'n > 0 . H 1/p o < 1 + 1/p, the operator I�+ is bounded from Lp ([a, b), p)
note
). mto H0
. .
([a, b); p), where
{
min(a -
if
1-'
-e
if
1-'
1/p, �-£/p), + �-£/p, with 1-' min l-'k t > 0 (KMapetyants and Rubin [2]) . k 2::2 ' �=
=
<
l .< oo , is bounded from Lp (a, b) into X'Y (a, b) if "Y
� 1/p' and unbounded if "Y < 1/p' . The proof is based on the fact that = O(r 11P' ) as r - oo. We note that BMO(a, b) (/. X-y (a, b) and X-y (a, b) (/. BMO(a, b) for 0 "Y 1. The first is = + o(1 ) ) as r - oo in the case attested by the example giving the relation = a = 0, b = 1. The example corresponding to the second assertion is given by 1
a, x $ a.
4. If 0 < Rea < ReI' the equations
[
1+a (1 f(J.t) ± ix) �' I-a (xf(J.t) ± i) �'
[
] ]
f(J.t - a) - e± �
(5.24)
=
(5.25)
_
(1 ± ix)�£-a ' f(J.t - a) (x ± i)�£ -a '
are valid with the power functions ( 1 ± ix )�', ( x ± i)�' being understood in the normal way, i.e. as corresponding values of the main branch of the analytic function z�' in the complex plane with the cut along the positive half-axis z
JA - I z I �' e i�£ arg z , e -1'1m+O arg z I z=t +ie ' t>O -- 0 . -
(5.26)
By choosing (5.26) we may write that
(±ix + 1) �' = (1 + x2 ) � e ±i�£arctg x , x E R1 , (x ± i)�' = (1 + x2 ) � e ±i�£arcctg x , X E R1 .
(5.27)
Equation (5.24) may be rewritten as (5.28) We find it convenient to prove equations (5.24) and (5.25) later, at the end of § 7.1. For the moment we remark only that these formulae are deducible one from each other in view of the connection given in (5.9) , provided that we take into account that (1 ± ix)�' l x = - e = e� iwrr l 2 (e ± i)�' by (5.27). Thus it will be sufficient to prove only one of the equations, either (5.24) or (5.25).
5.2. Fractional integrals of Holderian functions Results of this subsection are similar to those of § 3, the difference being in the specialness due to the presence of infinity on the axis R1 or half-axis. We start
§ 5. THE MAIN PROPERI'IES OF FRACTIONAL INTEGRALS
with the case of weighted Holderian functions on the half-axis consider fractional integrals
a 'P -_ _1_ zO+ f( a) Ot J_ rp -
1_ _ r (a)
R� = [0, oo] .
- t) 1 - 01 ' J (zrp(t)dt
99
We
:&
0
J
00
:&
(5.29)
rp(t)dt , z > o. (t - z) l -OI
The statement about their Holderian property on the half-axis will be obtained by reducing it to the case of a finite interval via the following lemma.
The transformation y = 1/(z + 1) maps the space H>. (R� ; p), p = p(z), z > 0, onto the space H >. ([o, 1]; r), where Lemma 5.1.
r = r(y) = p[(1 - y)fy],
o
< y < 1.
(5.30)
The lemma's proof may be obtained by direct verification. Note that the change of variable y = 1/(z + 1) transforms the Holderian condition (1.6) into
(1.1).
Theorem 5.1.
Let rp(z) E HSCR�; p), where
p(z) = (1 + z)11 IJ lz - Zk l "" , 0 = Z 1 < Z2 < · · · < Zn < oo n
k= 1
k=
Let also A + a < 1 and A + a < J.lk < A + 1,
(5.31)
2, 3, . . . , n. If
n
1' 1 < A + 1,
JJ
+ L: J.lk < 1 - A, k =1
(5.32)
then IC+ 'P E H� +01 {1l� ; p*). lfA+a < 1' 1 < A+ 1 {or p 1 = 0) or a-A < p+ kE J.lk , +1 then I�rp E H�+ 01 (R�; p*). Here p*(z) = (1 + z) - 201p(z) in both cases. Proof. Theorem 5.1 is reduced to Theorem
3.4 by changing variables. Indeed the
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
100
substitutions y =
1/(z + 1), r = 1/(t + 1) lead to
J z
0
J
00
z
1 tp (!:;t-) d r _0 1 - 11 r1+ cr ( r - y) 1 - a '
(5.33)
1 ) dr tp(t)dt = 1 _ 0 v tp ( 7' Y r1+ a ( y r) 1- cr · (t z p-er
(5.34)
tp(t)dt (z - t) 1 - cr
J
_
'
J
_
_
0
By Lemma 5.1 the change y = n>-([o, 1]; r(y)) with weight
1/(z + 1) maps the space n>-ck� ; p) onto the space n
r( y) = p[(1 - y) fy] = c IT IY - Ylc IJA" , k :O
(5.35)
n E J.lk , and Yk = (1 + z�c) - 1 , k = 1, 2, . . . , n. Applying Yo = 0, JJo = -JJ - lc= 1 Theorem 3.4 to the integrals (5.33) and (5.34) we obtain the proof of the theorem
where
•
after simple transformations.
Corollary 1. In the case p( z) = z" ( 1 + z ) �A the operator
n; +a (R�; p• ), � + a < 1, p*(z) = x" (1 + x)JJ - 2Q if v
0, � + a < 1 with �(x) E n>. + a ([o, oo]), �(0) = �(oo) = 0. Now we give a theorem for fractional integrals on the whole real line which is similar to Theorem 5.1 but essentially different due to other conditions on the parameters. Theorem 5.2
.•
Let cp(x) E H$( R1 ; p), where n
p(x) = (1 + x 2 ) t IT lx - x�: l#'" , -oo < Xt < A:=l
· · ·
< Xn < oo.
(5.39)
If � + a < 1, � + a < J.'A: < � + 1, k = 1, . . . , n, and a - � < I' + E J.'A: < 1 - �' A:=l then If.
0. Then the left hand side belongs to H� + 01 (R� ; p* (x)) by Lemma 5. 1. • Remark 5.1. Some results on mapping properties of fractional integrals in the
spaces of smooth functions on the axis or half-axis may be found in § 8.2, 8.4.
5.3. Fractional integrals of summable functions We consider here fractional integrals of functions cp E Lp , given on the axis or half-axis. The difference from the case of a finite interval is the following. In the case of a finite interval the fractional integration operators were defined (see § 3. 3 ) on any space Lp , 1 5 p $ oo, and mapped Lp , 1 < p < oo, into L9 with any q such that 1 5 q 5 p/(1 - crp) when crp < 1 and 1 $ q < oo when crp � 1. But for the case of the axis or half-axis these operators are well defined for 1 5 p < 1/cr and may map Lp into £9 for 1 < p < 1/cr and q = p(1 - crp) only. More exactly, the Hardy-Littlewood Theorem 3.5 in the case of the whole axis holds in the following form.
§ 5. THE MAIN PROPERI'IES OF FRACTIONAL INTEGRALS
103
Let 1 � p � oo, 1 � q � oo, a > 0. Operators I± are bounded from Lp (R1 ) to L9(R1 ) if and only if O < a < 1, 1 < p < 1/a and q = p/{1 - ap).
Theorem 5.3.
We omit the proof of this theorem as well as that of Theorem 3.5 (see references in § 9) and demonstrate here only the simple proof of the necessity of conditions a E {0, 1), p E {1, 1/a), q = p/{1 - ap). Let II I+ cpll 9 � cllcpllp · Then II I.�ll.s cpll 9 � cll ll.s cpllp as well, ll.s being the operator (5.11 ) . By (5.13) and the equality II IT .s cpllp = 6 - 1 /P II cpllp we obtain II I+ cpll 9 � cc5 a + t - � llcpllp · Letting c5 to tend to 0 and to oo, we note that this inequality may be valid only for 1/q = 1/p - a. Since q > 0 we have p < 1/ a. It remains to exclude the case p = 1. The function
cp( z) =
{1
"i
0,
I n_..., 1 , "i
0 < z < !, > 1, z ¢ (0, 1/2), 1
(5.41)
is an example of a function in L 1 (R1 ) such that (I+cp)(z) ¢ L 1/(1 - a) (R1 ) if 1 < 1 < 2 - a. Indeed for 0 < z < 1/2 we have
(5.42) so I+cp E L 1 /( 1 -a) (R1 ) only when {1 - 1)/(2 - a) > 1, i.e. 1 > 2 - a. It is evident that Theorem 5.3 is valid also for fractional integrals (5.29) on the half-axis (0, oo) . Some information for the case p = 1 may be found in Theorem 5.6. We give the following weighted variant of Theorem 5.3 for the case of the half-axis. Theorem 5.4.
Let 1 � p < oo,
1 0 < a < m + p-,
0 � m � a,
q = 1 - (ap- m)p
(5.43)
with m � 0 for p = 1. Then operators 1�, IC+ are bounded from Lp (R� ; z") into L9(R� ; z"), 11 = (pfp - m)q:
{ I00
z" lecp)( z) l 9 dz
} { I00 1/ q
S K
zPicp(z) i"dz
}
1 /p
(5.44)
104
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
(similarly for IC+ cp), where p < p - 1 /or the operator 10+ and p > Qp - 1 /or the operator I� . The statement of Theorem 5.4 for operator IC+ was earlier proved in Theorem 3.7 and it may be obtained for I� from its validity for 10+ by substituting 1/z = y, y- l - a cp(z) = 'Pt(Y), Pt = -p + p + Qp - 2, lit = -11 - 2 + (1 - Q)q in view of the equality (IC+ cp) (z) = y 1-a ( I�cpt )(y). Let us note the important particular cases of Theorem 5.4. The case p = 0, m = 0 leads to Theorem 5.3, while the case p = 0, m = Q gives the inequalities K
00
00
j zaP I(I�cp)(z) IPdz � P j jcp(z) jPdz, 0
1 � p < 1/ Q,
00
0
0
< Q < 1; KP
00
j zaP I (I0+ cp)(z) IP dz � j jcp(z) jPdz, 0
(5.45)
0
(5.46)
1 < p < oo, Q > 0, known as Hardy inequalitie.�. One may also obtain these inequalities independently of Theorem 5.4 by using Theorem 1.5 by limiting oneself to the condition 1 < p < 1/Q in (5.45). Theorem 1.5 gives herein the value of the constant K: K=r K= r
G ) / G) (�) jr ( �) -o
r
·
o+
for the inequalities (5.45) and (5.46) respectively. It is not difficult t o show that these constants are sharp. We find it convenient to single out the case m = Q (i.e. the case q = p) in Theorem 5.4 by rephrasing it (after the notational change pfp = - '"( , cp(z) = z7 /(z)) in the following way. The operators zP I�z-r and zfJ IC+ z-r, Q 0, are bounded from Lp(R�) into
>
105
§ 5. THE MAIN PROPERTIES OF FRACTIONAL INTEGRALS
L, (R� ; z -.P(a+.8 +"Y ) ), p � 00
J 0
1, if (a + ;)p < 1 and (; + 1)p > 1 respectively:
z -(a+.8 +"Y ).P i z.8 I�z"Y /(z)l'dz :5 K.P
J 1/(z)l'dz, 00
(5.45')
0
1 :5 p < oo , (a + ;)p < 1, a > 0;
00
00
j z-(a+.8+"Y).P iz.8I0+ z"Y /(z)l'dz ::; K' j 1 /(z)l'dz, 0
(5.46')
0
1 :5 p < oo , (; + 1)p > 1, a > 0.
In particular, the operators z.8 I�z"Y and z.8 IC+ z"Y are bounded in L ( R�) in the case a + {J + ; = 0 under the above conditions. The inequalities (5.45') and (5.46') with z-pRe (a +P +"Y ) instead of z -(a +,8 +-y )p are valid for complex values of a, {J, ; also if pRe(a + ; ) < 1 and p(Re; + 1) > 1 respectively, in both the following cases Rea > 0, 1 :5 p < oo or Rea = 0, a =I 0, 1 < p < oo . We observe that the purely imaginary case a = i(J may be considered by representing z -"Y ��� z"Y in the form ��� + (z - "Y ��� z"Y - ��� ) where the first term may be treated by Lemma 8.2 and the second one by Theorem 1.5. In the bounding cases of restrictions to inequalities (5.45' ) and (5.46) when p = 1 and ; = 1 - a or ; 0 respectively, the integrals on the left-hand side of these inequalities may diverge. In these cases we arrive at the inequalities (3.17"') and (3.17") instead of (5.45') and (5.46'). The inequalities (3.17"') and (3.17") with A = 0 show in particular that the operator z - 1 J�z l - a is bounded from L 1 ((b, oo) ; In !:p-z) into L 1 (b, oo) , and z -a 10+ is bounded from £ 1 ((0, b); In �) into £ 1 (0, b), 0 < b < oo . We give now without proof the generalization of Theorem 5.4 for the case of a general power weight (5.39). We consider z E 0 where 0 may be the half-axis R� ,
=
or axis R1 . In the case 0 � R� we set
0 = Z l < • · · < Zn
0.
(5.47)
106
Let JJo =
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
-p -
JJ I - · · · - JJn · We also let
11�) =
_
p1 q
P
_
t
k =2
liJc
_
{ eJJoq/- qpfp'
for JJo > 1 - p, for JJo ::; 1 - p, e >
0,
{
for JJo > 1 - p, for JJo ::; 1 - p , e >
0,
JJoq /p 1100(2) = -mq - � L...J liJc e - qfp' k= l
in order to define weighted functions of the type
_
r_ (z) -
{
(1 + z) "i!>
r+ (z)
(5.39):
ll lz - z�c l""
k= l
.
for for
0 = R�, 0 = R1 .
Let 1 < p < oo, 0 ::; m ::; a, 0 < a < m + 1/p, and let p(z) be the weight (5.39) satisfying condition (5.47) in the case of the half-line. Let
Theorem 5.5.
JJ k < p - 1 ,
k = 2, 3, . . . , n .
(5.48)
If in addition to (5.48) we also have JJ l < p - 1, then the operator Ig+ is bounded from Lp (R� , p) into L9(R� , r+ ) · If in addition to (5.48) we have that JJ < 1 - ap, then I� is bounded from Lp (R� , p) into L9(R� , r- ). Finally the operators I± are bounded from Lp (R1 , p) into L 9 (R1 , r± ) if in addition to (5.48) we have that JJ 1 < p - 1, JJo < 1 - ap.
§ 5. THE MAIN PROPERI'IES OF FRACTIONAL INTEGRALS
107
We note the particularly useful case of Theorem 5.5: �
{_l lzi" I(I.+ The case /(z) = IC+ cp is treated similarly, the only difference being that we deal with the integral
A leads to (5.51).
/t 1 we shall consider them in § 5.6 below.
5.5� The finite part of integrals due to Hadamard Comparing the Marchand fractional derivative D ± f =
r(l�a) J J (�1R:J(�) dt with 00
0
fractional integrals I±f, we see that D ± f is formally obtained from l±f if we replace a by - a . Subtraction of f(x) here provides the convergence of the integral. Thus D ± f are closely connected with ideas concerning divergent integrals. We elaborate on some of these ideas. Definition 5.1. Let a function 4>(t) be integrable on an interval c < t < A for any A > 0 and 0 < € < A. The function 4>(t) is said to possess the Hadamard property at the point t = 0 if there exist constants ak , b and A k > 0 such that A
N
j 4>(t)dt = L
k =l
�
a k €-).,.
+ b ln � + J0 (c) ,
{5.64)
where e-o lim J0 {c) exists and is finite. By definition A
p.f.
j 4>(t)dt
= lim Jo (c) . e-o
0
(5.65)
The limit (5.65) is called a finite part (partie finie) of the divergent integral J 4>(t)dt in the Hadamard sense or simply an integral in the Hadamard sense. The 0 constructive realization of the function J0 (c) is sometimes called a regularization A of the integral J 4>(t)dt. 0 It is not difficult to see that constants a k , b, A k in (5.64) do not depend on A. If 4>(t) is integrable at infinity, by definition we put A
p.f.
A
J 4>(t)dt = p.f. J 4>(t)dt + J 4>(t)dt oo
0
0
oo
A
(5.66)
113
§ 5. THE MAIN PROPERTIES OF FRACTIONAL INTEGRALS
and it easy to see that this definition does not depend on the choice of .
Now we return to
Di. / and consider the divergent
integral
next lemma holds.
A.
J J(:i+2dt . 00
0
The
Let 0 < a < 1 and let /(z) be locally Holderian of order ..\ > a . Then the function c)(t) = /(z - t)t - 1 -a possesses the Hadamard property at the point t = 0 for each z and if 1/(t)l :::;; cltl a-�, e > 0, as t --+ -oo, then �ma 5.2.
f
00
00
f(z - t) dt = f(z - t) - /(z) dt. p. . J t 1+a J t 1+a 0
The
0
proof of this lemma may be obtained by direct verification of condition (5.64)
and definitions (5.65) and (5.66) . Lemma 5.2 states that
D±f = p.f.l± a /,
0 < a < 1.
(5.67)
One may also say that (D±f)(z) represents for any z the analytic continuation of the function (l± a /)(z) from the half-plane Also this continuation is extended to the half-plane < for the functions /(z) mentioned in Lemma 5.2. This follows from the analyticity of the functions c) 1 (a) = If} = l'±a f in the half-planes and > respectively (for sufficiently "good" functions and from the coincidence of their boundary values: = limo c) ( ) = .P£m a f. limo
Rea ..\
�2 (a) Rea- + �t( a)
Re a- The conclusion D�+ /
/) 2a
Re a < 0.
Re a 0, Re a < 0
0 < a < 1 , similar to (5.67) is valid also for
= p.f.IO: /, the Marchand fractional derivative (5.63).
The interpretation of D±f in (5.67) indicates the way how one may make Marchand derivatives meaningful for � 1. For this purpose we give the a regularization of the divergent integral I+ in the next lemma.
a
/, a > 0,
Let locally /(z) E em and let f(m) (z ) satisfy locally the Holder condition of order .\ , 0 ::=;; ..\ < 1 . Then the function � (t) f(z - t)t - 1 - a possesses the Hadamard property at the point t 0 for any z if Re a < m + ..\. If also
Lemma 5.3.
=
=
CHAPTER 2 . FRACTIONAL INTEGRALS O N THE REAL AXIS
114
1 / (t ) l � clt l a- e
for t -+ -oo, then
m 1 ( -1)A: it· j(A:) (z) t) f(z E j t ) 1 1 j ( x f A:=O = p.f. dt r(-a) t 1 + ct dt r(-a) t 1+a oo
•
0
0
00
1 j f(x - t) dt + + -r(-a)
0
t1+a
m ( - 1) A: --
�
A:- 0
k!
(5.68)
f( A:) (z)
r(-a)(k - a) '
where Re a < m + �' a =/= 0, 1 , 2, . . . The
proof of this lemma is obtained by direct verification.
m = [a] the equality (5.68) may be rewritten as
We note that after the choice follows
. f(x - t) - E < -A:t') f(A:) (z) j f(x t) 1 1 A:=O · p.f. dt, r(-a) t 1 +a dt = r ( - a ) j t 1 +ct oo
oo
0
(a]
•
0
a =I= 0 , 1 , 2, . . . ,
(5.68')
where the integral on the right-hand side converges absolutely for the functions mentioned in Lemma 5.3. Result (5.68') has an advantage over (5.68) in being more compact. In view of (5.67) and of the analyticity of the right-hand side in (5.68) with respect to a, it is natural to use (5.68) for defining the fractional derivative of order a, Re a > 0. Let us show that such a definition agrees well with the definition (5.7) in the case of sufficiently "good" functions f.
Let f(x) satisfy the assumptions of Lemma 5.3 with m � [a] + 1 . Then the Liouville fractional derivative Vi.! coincides with (5.68) for any a such that Rea > 0, a =/= 1, 2, . . .
Theorem 5.8.
Proof. Let /3 = a - n + 1, n = [a] + 1 (0 < /3 < 1) in correspondence with For the "truncated" Liouville derivative we have z- e
f
dx" .!!_ -oo
f(t)dt = ( (z - t )P
_
1 )" (/3) n -
f
00
e
(5.7).
- 1 ( -1 ) 1c (/3) �: (n 1 lc f(z - t)dt + n""" + tn P LJ cA:+P / - - ) (z c ) ' A: =O _
(5.69)
1 15
§ 5. THE MAIN PROPERTIES OF FRACTIONAL INTEGRALS
which may be .proved by direct differentiation of the left-hand side integral. Regularizing the right-hand side integral a.s in (5.68), we obtain
oo /(z - t)dt /1 [/(z - t) - � (-1)k (t - e)k/(k) (z - e)] ..!!!___ tn+P J tn+P =O kLJ =
k!
e
e
(5.70)
with the designation
1
a�:(e) = eP+ k J (t - e )"- 1 - k t - n - P dt = £
. 1-•
J en- 1- k (e + 1)-n-Pde. 0
Substituting (5.70) into (5.69) we arrive at the equality
oo J
- t ) dt + n�1 ( -1) k /(n - k - 1) (z - e + f(z LJ tn+P k = O e k +P 1
) [(a)k - (n
. 00 00 We have a�:(e) = f en - 1 - 1: (1 + e) - n - P c.te - f 0
. 1 -•
JJ
(/3),. a�: e - k - 1) I ( ·
en - k - 1 (1 + e) - n - P de.
>] } .
Here the first
integral is easily reduced to the beta function. Changing the variable e + 1 in the second integral, we obtain
So the second square bracket in
(5.71) is equal to
(5.71) =
1/et
116
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Then passage to the limit in (5.71) as e --+ 0 is easily realized. So we have that the Liouville fractional derivative
indeed coincides with the right-hand side of (5.68) taking into account that ( - 1 )n (,B)n = r( n - ct ) /f( ct) by (1.46) and (1.47) . • -
5.6. Properties of finite differences and Marchaud fractional derivatives of order a > 1 Equations (5.57) and (5.58), defining Marchaud fractional derivatives, may be extended to the case > 1. One of the ways which comes to mind is to procede in a similar fashion to (5.7) by putting and introduce
a
a = n + {a }, n = [a]
It is possible however to choose another way by in� roducing differences of higher order, that is, I > 1 in (5.57) and (5.58) instead of the first order difference. We shall elaborate on this latter way. It will be preferable in some aspects because it shows directly the analytic dependence of D+f on the parameter ct. Firstly we consider some simple properties of finite differences. In terms of the translation Th we introduce
( dL /)(z) (E - TJ. )1 f =
which is said to be a finite
�(-1)1 G) /(z - kh), I
=
(5.72)
difference of order I of a function /(z) with a step h
and with center at the point z .
We shall need the following function of the parameter
a: (5.73)
It arises as a finite difference of the power function:
(� �/)(0)
=
-A,(ct) for
117
§ 5. THE MAIN PROPERI'IES OF FRACTIONAL INTEGRALS
/(z) = l z l a . The following property of this function:
A,(a) = 0 for a = 1, 2, . . . , 1 - 1 ,
(5.74)
will be important for us. It follows from the obvious equality
A1(m) = -
( !r (1 - z)'l�=l '
(5.74' )
•
As for non-integer values of a, it may be shown that A 1 (a) -:/; 0 for 1, see below and Lemma in Chapter 5. a -:/;
1, 2 , . . . , 1 -
a
(5.8 1) 26.1 Lemma 5.4. Let /(z) E cm (R1 ) and let I � m. Then 1 m h - k km (k1 ) J<m> (z - khu)du. ! m (-1) (�� /)(z) = (m - 1)! (1 - u)m- 1 � LJ k=O O I
E
R1 ,
(5.75)
Proof. By Taylor's expansion (with the remainder in the integral form) we have
1 ( -kh)m (h) i ) ( /(z - kh) E �, ' / (z) + (m - 1).l 1(1 - u)m - 1 /(m) (z - khu)du. =
. _0
0
a.
·-
So for the differences (5.72) we obtain the equality
which yields (5.75) in view of (5.74). Corollary 1.
If f(z)
E
cm (R1 ) and
•
J<m>(z) is bounded, then
1 ( ��/)( z)l � clhlm
sup 1/(m ) (z)l, 1:
I�
m,
(5.76)
116
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
Then passage to the limit in (5.71) as e --+ is easily realized. So we have that the Liouville fractional derivative
0
00
'D+f = r(n � a) ::n J f(:e - t ) cfl dt 0
(-1)n (P) n = r(n - a)/r( - a) by (1 .46) and (1 .47). •
indeed coincides with the right-hand side of (5.68) taking into account that
5.6. Properties of finite differences and Marchaud fractional derivatives of order a > 1 Equations (5.57) and (5.58), defining Marchaud fractional derivatives, may be extended to the case a > 1. One of the ways which comes to mind is to procede in a similar fashion to (5.7) by putting a = n + {a}, n = [a] and introduce
n f(n ) (X) - f(n ) (X - t ) dt. D+ f - dxd n D {+0} I - r(l{a} J t l + { a} - {a} 0 0
_
00
_
It is possible however to choose another way by in� roducing differences of higher order, that is, I > 1 in (5.57) and (5.58) instead of the first order difference. We shall elaborate on this latter way. It will be preferable in some aspects because it shows directly the analytic dependence of D i./ on the parameter a. Firstly we consider some simple properties of finite differences. In terms of the translation Th we introduce
( .O.L f)(z) = (E - TJ. )1! =
�( - l)i G) /(z - kh), l
(5.72)
which is said to be a finite
difference of order I of a function f(x) with a step h and with center at the point x. We shall need the following function of the parameter a: (5.73)
It arises as a finite difference of the power function:
(�i/)(0)
=
-A,(a)
for
§ 5. THE MAIN PROPERTIES OF FRACTIONAL INTEGRALS
117
f(x) = l z lcr . The following property of this function: A,(a) = 0 for a = 1 , 2,
. . . ,
1 1 -
,
(5.74)
will be important for us. It follows from the obvious equality
( )
d m (1 - x) ' , A1 (m) = - x dx l�= 1 . As for non-integer values of a, it may be shown that A 1 (a) :f= 0 for a :f= 1 , 2, . . . , 1 -J, see (5.81) below and Lemma 26.1 in Chapter 5. Lemma 5.4.
(5.74')
a E R1 ,
Let f(x) E cm (R1 ) and let I � m. Then
Proof. By Taylor's expansion (with the remainder in the integral form) we have
1 m m m - 1 ( kh)i m ) ( kh ( ) i 1 ( 1 / (x) + f(x - kh) = L -T-(m _ 1) .1 (1 - u) - / ) (x - khu)du . •. = 0 a. 0 So for the differences (5.72) we obtain the equality
( �� /)(x)
m-1 ( L -.�)' /(i)(z)A,(i) i:O ·
= -
I.
which yields (5.75) in view of (5. 74). Corollary 1.
•
If f(x) E cm (R1 ) and f(m > (x) is bounded, then 1 ( �� /)(x) l � clhlm sup IJ<m> (x) l , l � m, �
(5.76)
120
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
5.7. C onnection with fractional powers of operators The operator 'Di. of fractional differentiation may ·be considered as a fractional power of the differentiation operator: (5.82) under the appropriate interpretation of fractional power of an operator. In fact this was the main model in mind in the development of the abstract theory of fractional powers in Banach spaces. We refer the reader who wishes to become more familiar with this theory to the books by Krasnosel'skii, Zabreiko et al. [1) and by Yosida [1]. We mention only very briefly the simplest definitions in this theory and show that they include the case of fractional integro-differentiation in a suitable setting. Let X be a Banach space and let {T, } , t � 0, be a strongly continuous semigroup in X (see Definition 2.5). The operator
A = t -O+ lim !(r, - E) t
(5.83)
( X)
is said to be a generator (or infinitesimal operator) of the semigroup Tt . It is known (see e.g. Dunford and Schwartz's book [1], p.660) that the domain D(A) of the operator A is dense in X and that A is a closed operator. The equality Tt = e tA is valid, at the least formally, the exact meaning being T, = lim e tA" , h- o Ah = t (Th - E)) . We shall consider fractional powers ( -A) a for operators A, which are generators of strongly continuous semigroups. A positive power of an operator A is defined by the formula 00
( -A) a 'P = r ( �Ot) J c a- '(T,
0 and from (5.86) we have
122
CHAPTER
2. FRACTIONAL INTEGRALS ON THE REAL AXIS
where cp E Lp,w 1 ::; p ::; oo , w > 0, and the above mentioned Lp,w (at infinity) of the integral is implied. As for (5.84), we obtain
(.!!_dx_)
Q'
00
1 _ cp(x - t) - 2 and to f(x) 0 for lxl < 2. The imbeddings (6.2), (6.3) together with (6.4) mean that
= l q
=
(6.5)
So,
in view of Theorem 5.3, the space Ia (Lp ) does not coincide with any space Lr ( R 1 ), 1 5 r 5 oo. It does not coincide with any space Lr ( R1 ; p), either. Therefore the space I a (Lp ) needs to be characterized. Subsection § 6.3 is devoted to this purpose. Firstly we consider the inversion of fractional integrals I+ 1.
(6. 10)
We note that k (e) E L 1 (R�) and that
(6.1 1)
§ 6. REPRESENTATION BY FRACTIONAL INTEGRALS OF Lp-FUNCTIONS
In view of (6.9) we obtain the relations
125
oo oo
dt k ( e ) cp(x - e) (D +,e f)(x) = r (1 a- a) j t2 cJe j t £ Q
0
00 F./ £ a cp(x e = r(1 - a) j e cJe j k(s)ds 0 0
00
t
= r (1 a- a) j cp( x t- ct) dt j k(s)ds. 0 0 Here t
j k(s)ds = a - 1 t r(1 - a)K(t)
(6.12)
0
which may be proved by direct evaluation of the left-hand side. Then ( 6.8) becomes evident. • We note that in a generalization of ( 6.6 ) the representation 00
(D t.,e f)(x) = j Kl, a (t)cp(x - ct)dt
(6.6' )
0
may be similarly obtained for arbitrary a > 0, where the truncated Marchand derivative ( 5.80' ) is used in the left-hand side and I
E (-1)l: (!) (t - k)t.
Kl , a (t ) = 1::0x(a, l) r (1 + a)t
(6.7' )
It is not difficult to show that
K1, a (t) E L 1 (R1 ) Theorem 6.1.
J Kl, a (t)dt = 1. 00
and
(6.8' )
0
Let f(x) = l±cp,
0. Let us denote
(A h 1, ; + � = 1 + o: and ! - ! _ OA ,., ! - ! _ OA ,., o I t •
r
fndeed, these assumptions being satisfied, then I E JOt(Lp), g E JOt(Lr) and
therefore (6.27) follows from (5.16). In order to formulate another corollary let us introduce the space of functions in Lr(R1 ) , which have fractional (Marchand) derivative in Lp(R1 ) : (6.28)
Corollary 3.
Let 0 < o: < 1,
1 < p < 1/o:, 1 � r < oo.
Then (6.29)
Remark 6.1.
In view of (5.49) the following weighted variant of Theorem
6.2 is valid: a function /( z ) is representable by a fractional integral I+cp with
130
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
cp E L,(R1 ; l z l �'), ap - 1
1 , if and only if
1 1 < -1 . CR 1 ),
$ K ln(2 + lzl),
•
A > a, then
l( D + ,e f)(z) l $ c( 1 + j z 1 ) - >. - o ,
a < A < 1,
I ( D + ,e f)(z ) l $ c(1 + jz 1 ) - >. - o ln( 2 + lzl), A = 1, where c does not depend on z and e. If herein A > max( a, - a + 1/p) then f(z) E l0(Lp)·
(6.33) (6.34)
and /(oo) = 0,
Proof. In view of the Holderian condition ( 1.6) on R1 we obtain the inequality I ( D + f )( z )l :5 .•
1 (1 + izi) �
f tl+a- � (1� iz - ti)� ' 00
0
Applying here (6.32), we obtain (6.33) and (6.34) . If /(oo)
=
0 and A >
-o
+ 1/p,
§ 6. REPRESENTATION BY FRACTIONAL INTEGRALS OF Lp -FUNCTIONS
133
then I E £ 9 , 1/q = -a + 1/p. Furthermore it follows from (6.33) , (6.34) that sup IID+,e iiiP < oo. Then I E I ( Lp ) by Theorem 6.2. • . t >O The next theorem gives sufficiency conditions in the weighted terms.
01
Theorem 6.6.
If l(:x) = lxl,. fi:1xD" , where g(:x) E H>.. C R1 ) , then (6.35)
where A > a, - a < JJ � 1, v > a and c doesn't depend on :x and c. In the case = 1 one more factor ln(2 + l:xl) is needed in (6.35). If besides
v + JJ
1 q
1
-1 = -1 - a,
-,
- - v < JJ
Po ·
Theorem 7.3.
=
(Ig+ !p)(x) = (d/dx)"(IgJ"!p)(x), Rea + n > 0, according to (2.32), we, first, apply ( 1.124) taking into consideration that (d/dx)i (IgJ" �P)(x) = 0 with z = 0 and j = 0, 1, 2, . . . , n - 1 follow from the conditions IP(j ) (O) = 0, j = 0, 1, 2, . . . , n - 1. Then we use Theorem 7.2 with respect to the integral IgJ "
0, o
(7.21)
a X - a tp( X )) * ( S ) = r (l - a -) s) tp* (S) ' Re(a + s) < 1, (LO+ f( 1 - S
(7.22)
which after the substitution f(x) = x - a rp(x), according to (1.117) take the form
Res < 1.
(7.23)
The conditions for when these formulae are valid are contained in the following theorems.
Let Rea > 0 and f(t)t8 + a- 1 E L 1 (0, oo) . Then (7 .20) holds if Res < 1 - Rea while (7.21) holds if Res > 0.
Theorem 7.4. The
proof is carried out in the same way
conditions on
a and s
the proof of Theorem ensure the existence of the inner integrals. as
7 .2.
Given
Let -n < Rea � 1 - n, n = 1, 2, . . . , f(t) E cn ([O, b]), b being any positive number, and let f(t)t a+& - 1 E L 1 (0, oo ) . Then (7.20) holds if
Theorem 7.5.
§ 7. INTEGRAL TRANSFORMS OF FRACTIONAL INTEGRALS
1 43
Res < 1 - Re a and conditions X3 - k (Ig: k f)(x) = 0 for x = 0, x = oo,
k = 1 , 2, . . . , n,
(7.24)
are satisfied, while (7.21) holds if Res > 0 and the conditions x3 - k (J� + k f)(x) = 0 for x = 0, x = oo,
k = 1 , 2, . . . , n,
(7 .25)
are satisfied. Proof. By the condition f(t) E cn ([O, b]) the fractional derivative (Ig+ J)(x) exists. We write it down in the form (Ig+ J)(x) = d�".. (Ig:n f)(x) ( see (2.32)). Then we apply the Mellin transform to it and integrate by parts n times, thus
I
(I�+ f(z))*(s) = .,• - l d d�:�l u�r f}(z) 00
0
n-1 k =O
"" & k 1 O++ A: + 1 / (x) l xoo= O - L.J ( 1 - s) k x - - rr + ( 1 - s) n
I X3- n- 1 Ig:n f(x)dx. 00
0
By (7.24) the integrated terms are equal to zero. Applying Theorem 7.4 and having replaced a by a + n and s by s - n, respectively, we obtain the result
Hence, .we deduce ( 7.20) . The case of the integral I� is considered in a similar way. The theorem is thus proved. • Together with (7 .22) and (7 .23) the following theorem, characterizing the result of the application of fractional integrals and derivatives to the inverse Mellin transform, is used below as well. Theorem 7.6.
Let J• (s) E £2 ( 1/2 - ioo, 1/2 + ioo), 7J � min (O, Re (a - b)). Then
144
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
the equations X bla b X a -
O+
-
1 / 2 +ioo
J 1 / 2 -ioo
S j* ( S ) X - ds = 'I
6
1 / 2+ioo
f ( 1 - a - s) j*( ) X - ds, s) S
S f(l - b J 1 / 2 -ioo 'I
_
6
Rea < 1/2, Reb < 1/2;
:h�- • ., - •
j / 2 -ioo
1 / 2+ ioo
1
•"
(7.26)
��!: � f*(s)z - 'ds, j / 2 -ioo 1 / 2+ ioo
f*(s)z - ' ds =
•"
1
Rea > - 1/2, Reb > - 1/2.
(7.27)
hold. The proof follows from the existence of the integrals in (7 .26) and (7 .27) under
the formulated conditions on the parameters and function, from their absolute convergence almost everywhere and, as a result of this, from the possibility of interchanging the order of integration in the left-hand sides of (7.26) and (7.27) . After doing this the evaluation of the inner integral is carried out by { 1 .68). We also note that §§ 10, 18 and 36 contain various composition formulae reflecting the result of the application of some other integral transforms to the operators I� and I� , and their generalizations and modifications (see also §§ 9, 23 and 39).
= J t a-l f(x =f t)dt, Re a > 0, 0 are the Mellin transforms of the functions
0, J a1-ioo
(7.28)
with T > 0 and under the respective choice of signs. The formula may evidently be interpreted as an integral analogue of the Taylor series expansion. Further, taking in particular x = 0, we have
f( =t=r) = 2�i
a1+ioo
f(a)(I±f)(O)r - 0da, r > 0. J a1-ioo
(7.29)
§ 8. FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS
145
This means that / (z) may be restored by the values of its fractional integrals only at one point if the latter are known for all o on some line Reo = o 1 > 0.
(11/)(0)
§ 8 . Fractional Integrals and Derivatives of Generalized Functions We assume that the reader has some minimal knowledge concerning generalized functions. A generalized function is treated as a continuous functional on one or another space of test functions. Various such spaces are used depending on the problem in hand, in order to take into account the particular characteristics of the J>roblem. This will become obvious in the context of the present section.
� . 1 . Preliminary ideas We shall consider generalized functions over n, where n is the real axis or half-axis. Only § 8.5 will contain brief indications in the case of n being a finite interval. We cho_ose test functions on n to be infinitely differentiable at the interior points of n with prescribed behaviour at the endpoints of n. The value of the generalized function f as a functional on the test function cp will be denoted by ( / , cp ) , The generalized function is called regular if there exists a locally integrable function f(x) such that J f(x)cp(x)dx exists for each test function cp(z) and n
j
(/, cp) = f(x)cp(x)dx n
(8.1)
It is assumed that the bilinear form (/, cp) is chosen in such a way that it coincides with (8.1) in the case of a regular generalized function. The space X = X (O) of test functions is assumed to be a topological vector space. We denote by X' = X'(O) the topological dual space of X, i.e. the space of continuous linear functionals on X . Let us recall the notion of a generalized function concentrated at a point. A generalized function f E X' is said to be zero on an open set G, if ( /, cp) = 0 for each test function which is zero beyond G. The union 01 of all open sets where f = 0 is called a null set of the function f. The complement of the null set with respect to n is said to be the support of the generalized function and is denoted by supp I = n \ 0I · We say that the generalized function is concentrated at the point zo , if supp / is this point z0 • The well-known Dirac function 6 ( z - zo ), Z o E n, and its derivatives defined by
146
CHAPTER 2. FRACTIONAL INTEGRALS ON THE REAL AXIS
provide examples of generalized functions, concentrated at a point. The inverse statement is also true namely, any functional f, concentrated N at the point z0 , is of the form f = l: ck f5( k ) (z - zo) , - Vladimirov [2, p.52] or
k=O
Gel'fand and Shilov [1, p.149]. There are two main ways to define fractional integrals and derivatives of generalized functions. The first goes back to Schwartz [1] and is based on the definition of a fractional integral as a convolution (8.2) of the function rla ) z±-1 with generalized function f - see § 8.3. This way is well suited to the case of the half-line. The second way, which is more common, is based on using the adjoint operator. Namely, starting with (2.20) and (5.16) for fractional integration by parts one may introduce (8.3) by definition, 16_ , J� and fractional derivatives being defined similarly. The approach via (8.3) will be correct if 16_ continuously maps the space of test functions X into itself. Sometimes a more general treatment is admitted when f and 1:+ f are considered as generalized functions on different spaces of test functions X and Y so that f E X', 1:+ 1 E Y' and then If_ must map continuously Y into X. We shall outline (8.2) very briefly in § 8.3. The main attention is paid to (8.3). This is considered in § 8.2 in the case of the axis R1 while § 8.4 deals with the case of the half-axis. 8.2.
The case of the axis R1 • Lizorkin's space of test functions
The well-known space S of Schwartz test functions (which are infinitely differentiable and rapidly vanish at infinity together with all derivatives) as well as the class COO C S of finite infinitely differentiable functions is poorly adapted for fractional integrals and derivatives. It is obvious that functions I� 1 and applying the Taylor expansion with the remainder in integral form in the first case. X
Remark 8.1. Space � does not contain real-valued functions everywhere different from zero. This follows from (8.6) with k = 0. The space of linear continuous functionals on will be denoted by �' as is usual. Let us compare �' and S' . We begin by comparing lli' with S'. Since \li is closed in S, we may identify lli' with the quotient-space of the Schwartzian space
S'
§ 8. FRACTIONAL INTEGRALS OF GENERALIZED FUNCTIONS
151
\II� of functionals in S' having \II as a null space, i.e. = S'I\II� (8.9) where \II � = {/ : f E S', (/,1/J) = 0,1/J E \11 } . We have used the known general modulo the subspace
q,'
fact: namely if M is a closed subspace in a linear topological space E, then M' = E' IMl. , where Ml. is the space of all functionals in E', which are orthogonal to M. It follows from the definition of space that consists of functions concentrated at the point x 0. Then - see § 8.1 consists of linear combinations of the delta-function and its derivatives. It is well known that cS (A:) (z) is the Fourier transform of the power function, that is .1"{(-it)l: ; x } 27rc5(A:) (z), the Fourier transform being understood in the sense of generalized functions:
\II ,
=
\II� - \II�
=
(},
