The Mellin Transformation and Fuchsian Type Partial Differential Equations

232 Pages·1992·17.998 MB·English

by  Zofia Szmydt, Bogdan Ziemian (auth.)

The Mellin Transfonnation and
Fuchsian Type Partial Differential Equations
Mathematics and Its Applications ( East European Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. BIALY NICKI-BIRULA, Institute of Mathematics, Warsaw University, Poland
H. KURKE, Humboldt University, Berlin, Germany
J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia
L. LEINDLER, Bolyai Institute, Szeged, Hungary
L. LOVA sz, Bolyai Institute, Szeged, Hungary
D. S. MITRINOVIC, University ofB elgrade, Yugoslavia
S. ROLEWICZ, Polish Academy of Sciences, Warsaw, Poland
BL. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria
I. T. TODOROV, Bulgarian Academy of Sciences, Sofia, Bulgaria
H. TRIEBEL, University ofl ena, Germany
Volume 56
The Mellin Transformation
and Fuchsian Type
Partial Differential Equations
by
Zofia Szmydt
Department of Mathematics,
University ofWarsaw,
Warsaw, Pa/anii
and
Bogdan Ziemian
Institute ofM athematics,
Polish Academy of Sciences,
Warsaw,Po/anli
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Szmydt, Zofla.
The Mellin transformat ton and Fuchsian type partial dtfferential
equations I by Zofia Sz.ydt and Bogdan Ztemian.
p. cm. -- (Mathematics and its appltcatl0ns (Kluwer Academtc
Publishers). East European Series ; v. 56)
Includes tndex.
ISBN 978-94-010-5069-2 ISBN 978-94-011-2424-9 (eBook)
DOI 10.1007/978-94-011-2424-9
1. Dtfferential equations, Partial. 2. Melltn transform.
1. Ztemtan, Bogdan. II. Tttle. III. Title: Fuchsian type partial
dtfferenttal equations. IV. Sertes.
OA377.S969 1992
515' .353--dc20 92-4675
ISBN 978-94-010-5069-2
AlI Rights Reserved
@ 1992 Springer Science+Business Media Dordrecht
Origina1ly published by Kluwer Academic Publishers in 1992
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To the memory of our great friend and adviser
Professor Andrzej Plis
SERIES EDITOR'S PREFACE
'Et moi, .. Of si j'avail su comment en revenir. je One selVice mathematics has rendered the
n'y semis point alll!.' human race. It has put common sense back
Jules Verne when: it belongs, on the topmon shelf next to
the dusty canister labelled 'discarded nonsense'.
The series is divergent; therefore we may be Eric T. Bell
able to do something with iL
O. Heaviside
Mathematics is a tool for thought A highly necessary tool in a world where both feedback and nonlineari
ties abound, Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sci
ences,
Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser
vice topology has rendered mathematical physics .. , '; 'One service logic has rendered computer science
.. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements
obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have
appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and textbooks
on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and
related fields does not grow only by putting forth new branches. It also happens, quite often in
fact, that branches which were thought to be completely disparate are suddenly seen to be
related. Further, the kind and level of sophistication of mathematics applied in various sci
ences has changed drastically in recent years: measure theory is used (non-trivially) in
regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky
lemma. coding theory and the structure of water meet one another in packing and covering
theory; quantum fields, crystal defects and mathematical programming profit from homotopy
theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use
Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experi
mental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large
scale order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics. "
By and large, all this still applies today. It is still true that at first sight mathematics seems rather frag
mented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so
are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make
such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of interaction
areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots,
quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to
the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination
of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to
statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections
of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not
in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of
much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combina
torics, algebra, probability, and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the extra
viii
mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest
and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal
inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics
were linear we would have no fun with transistors and computers; we would have no TV; in fact you
would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and
anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical
engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the
shortest path between 'real' results. Similarly, the first two topics named have already provided a number
of 'wormhole' paths. There is no telling where all this is leading -fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five subseries:
white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still
applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are
used in others. Thus the series still aims at books dealing with:
a central concept which plays an important role in several different mathematical and/or scientific
specialization areas;
new applications of the results and ideas from one area of scientific endeavour into another;
influences which the results, problems and concepts of one field of enquiry have, and have had, on the
development of another.
The present volume in the series is a book about two things, maybe two and a half. The two are: the theory
of the Mellin transform, a very useful integral transform that, till now, has had no systematic treatment (in
more than one dimension), and Fuchsian type singular differential equations, the subject of Chapter III
which includes the authors' own important results. The half is an appendix on Ecalle's resurgent functions,
a most significant topic in my view, which can do with a few extra clear expositions here and there.
This is deep and up-to-date mathematics at the cutting edge of research, but, thanks to the authors, still
accessible to all those with a standard background. That, as one of my teachers once remarked, is a sign of
good research mathematics; within a few years of when they were obtained, the results should be explain
able to graduate students. All this gives me something like 3/1.2 reasons to welcome this volume in this
series, and I do so with pleasure.
The shortest path between two truths in the real Never lend books, for no one ever returns them;
domain passes through the complex domain. the only books I have in my library are books
J. Hadamard that other folk have lent me.
Anatole France
La physique ne nous domte pas seulement
l'occasion de resoudre des problemes .•. e\le The function of an expert is not to be more right
nous fait pressentir la solution. than other people, but to be wrong for more
H. Poincare sophisticated reasons.
David Butler
Bussum, 10 February 1992 Michiel Hazewinkel
CONTENTS
Series Editor's Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vll
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xiii
Chapter I. Introduction
§1. Terminology and notation ........................................... . 1
§2. Elementary facts on complex topological vector spaces ............... . 5
1. Multinormed complex vector spaces and their duals ............. . 5
2. Inductive and projective limits .................................. . 7
3. Subspaces. The Hahn-Banach theorem ......................... . 8
Exercise 9
§3. A review of basic facts in the theory of distributions ................. . 9
1. Spaces DK and (DK)' .......................................... . 9
2. Spaces D(A) and D'(A) .......................................... . 9
3. Spaces S and S' ................................................ . 15
4. Spaces E and E' ............................................... . 16
5. Substitution in distributions. Homogeneous distributions ........ . 16
6. Classical order of a distribution and extendibility theorems for
distributions ................................................... . 17
7. Convolution of distributions 19
8. Tensor product of distributions 20
Exercises 21
Chapter II. Mellin distributions and the Mellin transformation
§4. The Fourier and the Fourier-Mellin transformations ................. . 27
1. The Fourier transformation in S' ............................... . 27
2. The Fourier-Mellin transformation in the space of Mellin
distributions with support in R+. ............................... . 30
Exercises ........................................................... . 32
§5. The spaces of Mellin distributions with support in a polyinterval 34
1. Spaces Ma «0, t]) and M~ «0, t]) ............................... . 34
2. Spaces M(w) «0, t]) and M(w) «0, t]) ............................ . 41
Exercises 46
§6. Operations of multiplication and differentiation in the space
of Mellin distributions 48
IX
x CONTENTS
1. Multiplication and differentiation in M a, M(w) and their duals 48
2. Mellin multipliers .............................................. . 49
Exercises 52
§7. The Mellin transformation in the space of Mellin distributions ....... . 54
1. The Mellin transformation in the space of Mellin distributions
and its relations with the Fourier-Laplace transformation ....... . 55
2. Examples of Mellin transforms of some functions ................ . 60
3. Mellin transforms of certain cut-off functions .................... . 67
3.1. One-dimensional smooth cut-off functions .................. . 67
3.2. n-Dimensional smooth cut-off functions with a parameter 71
Exercises 73
§8. The structure of Mellin distributions 76
1. Characterizations of Mellin distributions .......................... 76
2. Substitution in a Mellin distribution 82
3. Mellin order of a Mellin distribution 86
Exercises 87
§9. Paley-Wiener type theorems for the Mellin transformation 87
Exercises 98
§1 0. Mellin transforms of cut-off functions (continued) . . . . . . . . . . . . . . . . . . . 100
1. Conical cut-off functions . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 100
2. The K -inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3. The "tangent cones" EK and related cut-off functions .......... 106
4. Further investigation of the Mellin transform of a conical cut-off
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Exercises 114
§11. Important subspaces of Mellin distributions . . . . . . . . . . . . . . . . . . . . . . . . . 115
1. Subspaces M(~) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2. Subspaces SPr(s, s') of Mellin distributions ..................... 119
3. Spaces M(nj e) and Zd(nj e) of distributions with continuous
radial asymptotics . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 121
Exercises 124
§12. The modified Cauchy transformation 125
1. Modified Cauchy and Hilbert transformations in dimension 1 125
2. The case with parameters ...................................... 128
Exercises 137
CONTENTS Xl
Chapter III. Fuchsian type singular operators
§13. Fuchsian type ordinary differential operators . . . . . . . . . . . . . . . . . . . . . . . . 139
1. Asymptotic expansions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 139
2. The equation P( x ddx)u = f and definition of ordinary Fuchsian
type differential operators . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 144
3. Case of smooth coefficients ..................................... 146
4. Case of analytic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
5. Special functions as generalized analytic functions . . . . . . . . . . . . . . 162
Exercises 174
§14. Elliptic Fuchsian type partial differential equations in spaces M(:) 175
1. Existence and regularity of solutions on tangent cones 5K 176
2. Case of a proper cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Exercise 187
§15. Fuchsian type partial differential equations in spaces with continuous
radial asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
1. The radial characteristic set Chaq P . . . . . . . . . . . . . . . . . . . . . . . . . . 190
2. Regularity of solutions in spaces M(n; e) and Zd(n; e) ......... 196
Appendix. Generalized smooth functions and theory of resurgent
functions of Jean Ecalle ................................................ . 205
1. Introduction 205
2. Generalized Taylor expansions ................................ . 206
3. Algebra of resurgent functions of Jean Ecalle .................. . 208
4. Applications .................................................. . 209
Bibliography ......................................................... . 215
List of Symbols ...................................................... . 219
Subject Index ........................................................ . 221