Table Of ContentApplied Hyperfunction Theory
Mathematics and Its Applications {Japanese Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
H. AKAIKE, Institute of Statistical Mathematics, Tokyo
T. KAWAI, Kyoto University, Kyoto
M. TODA, Prof Emeritus, Tokyo University of Education, Tokyo
Y. TAKAHASHI, University of Tokyo, Tokyo
Volume 8
Applied
Hyperfunction Theory
by
Isao Imai
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
1ma i, 1sao, 1914-
[Oyo chokansuron, Engl1sh)
Appl1ed hyperfunct10n theory I by 1sao Ima1.
p. cm. -- (Mathematics and its applications (~apanese series)
Translat10n of: Dyo Chokansuron.
Includes bibl10graph1cal references and index.
ISBN 978-94-010-5125-5
1: Hyperfunctions. I. Title. II. Series: Mathematics and 1ts
appl1cat10ns (Kluwer Academic Publishers). ~apane seser1es.
CA324.I4313 1991
515'.782--dc20 91-35799
ISBN 978-94-010-5125-5 ISBN 978-94-011-2548-2 (eBook)
DOI 10.1007/978-94-011-2548-2
All Rights Reserved
© 1992 Springer Science+Business Media Dordrecht
Originally 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.
SERIES EDITOR'S PREFACE
'Hl moi, ...• si favait so comment en revenir, One service mathematics has rendered the
je n'y scrais point aile:' human race. It has put common sense back
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non-
The series is divergent; therefore we may be sense',
able to do something with it. Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'e1:re 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 pUlling 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 sciences 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 thc 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 'experimental
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
fragmented 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, modu
lar 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 detcction 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 combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
vi
extra mathematical sophistication that this requires. For that is where the rewards arc. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; 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 fre
quently 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 sub
series: 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 subdis
cipline 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.
Mathematicians are rarely satisfied. And apart from that, the various applications also demand fre
quently that concepts are extended. Functions have been particularly vulnerable. Once, long ago,
they were formulas; then they became mappings; still not enough; distributions appeared; and,
now, in addition, physicists and engineers also have to learn to live with hyperfunctions. These con
stitute a formidable tool, which has amply proved its value, for instance in quantum field theory.
They are also formidable in terms of mathematical sophistication, and words like microlocalization,
sheafs, and cohomology tend to flutter around.
Fortunately, it is also possible to understand them in more down to earth terms, thanks to the
author of the present book, who devoted a decade of research to understanding them in terms that
are accessible to non-mathematicians, in particular engineers and fluid dynamicists.
Here is a detailed and complete account of this way of looking at things, and this volume brings
Sato's marvelous tool within reach of any mathematically reasonably trained scientists, including
those who shudder at the very word 'cohomology'.
The shortest path between two truths in the Never lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
1. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
l'occasion de resoudre des problemes ... eUe right than other people, but to be wrong for
nous fait pressentir la solution. more sophisticated reasons.
H. Poincare David Butler
Bussum, September 1991 Michicl Hazewinkel
Contents
Series Editor's Preface v
Preface xvii
Chapter 1. INTRODUCTION 1
§1 What is a hyperfunction? 1
§2 SaUl's hyperfunction 2
§3 Aim 3
§4 Complex velocity and analytic function 4
§5 Distribution of vortices and hyperfunctions 6
§6 Ordinary functions and hyperfunctions 8
Chapter 2. OPERATIONS ON HYPERFUNCTIONS 11
§1 Definition of hyperfunctions 11
§2 Linear combinations 13
§3 Product of a hyperfunction and an analytic function 14
§4 Reinterpretation of ordinary functions as hyperfunctions 15
§5 Differentiation of hyperfunctions 19
§6 Definite integrals of hyperfunctions 22
§7 Summary 23
viii
Chapter 3. BASIC HYPERFUNCTIONS 25
§1 Preliminary 25
§2 Hyperfunction with generating function F( -z) 25
§3 Even hyperfunctions and odd hyperfunctions 27
§4 Hyperfunction with generating function F(z) 29
§5 Real hyperfunctions and imaginary hyperfunctions 30
§6 Single-valued analytic functions reinterpreted as hyperfunctions 33
§7 Cauchy's principal value 34
§6 Hyperfunction of the form f(ax + b) 36
§9 Formal product of a hyperfunction and a single-valued analytic func-
M
t~n
§10 H(z), l(z), sgnz 40
§12 log lxi, log Ixl H(x), log Ixl sgnx 45
§14 x-m(log Ixl)n, x-m(log Ixl)n H(x), x-m(log Ixl)n sgnx 47
§15 Ixl<>, Ixl<>H(x), Ixl<>sgnx 47
§16 Equation ¢(x) . f(x) = h(x) 49
§17 Summary 51
Chapter 4. HYPERFUNCTIONS DEPENDING ON PARAMETERS 53
§ 1 Preliminary 53
§2 Hyperfunction depending on a parameter 53
§3 Operations on parameter dependent hyperfunctions 56
§4 Convergence of a sequence of functions and convergence of a sequence
of hyperfunctions 60
ix
§5 Ixla log lxi, Ixla log Ixl H(x), Ixla log Ixl sgnx 61
§6 Ixla(log Ixl)n, Ixla(log Ixl)n H(x), Ixla(log Ixl)n sgnx 64
§7 x-m(loglxl)n, x-m(loglxl)nH(x), x-m(loglxl)nsgnx 65
§8 Power-type hyperfunctions 67
§9 Finite part of a divergent integral 71
§10 Calculation of pf integrals 75
§11 Summary 81
Chapter 5. FOURIER TRANSFORMATION 83
§ 1 Preliminary 83
§2 Definition of Fourier transformations 83
§3 Theorems about Fourier transformation 88
§4 Inverse Fourier transformations 93
§5 Examples of calculations of Fourier transforms 97
§6 Summary 99
Chapter 6. FOURIER TRANSFORMATION OF POWER-TYPE HY-
PERFUNCTIONS 101
§1 Preliminary 101
101
§3 Flxla H(x), Flxla, Flxlasgnx 104
§4 Flxla(log Ixl)n H(x), Flxla(log Ixl)n, Flxla(log Ixl)n sgn x 105
§5 FxP(log Ixl)n H(x), FxP(log Ixl)n, FxP(log Ixl)n sgnx 107
§6 Fx-m(log Ixl)n H(x), Fx-m(log Ixl)n, Fx-m(log Ixl)nsgnx 109
§7 Table of Fourier transforms of power-type hyperfunctions 111
§8 Polygamma functions 111
x
§9 Examples of application 112
§10 Summary 114
Chapter 7. UPPER (LOWER)-TVPE HYPERFUNCTIONS 115
§1 Preliminary 115
§2 Left (right) hyperfunctions and upper (lower) hyperfunctions 115
§3 Properties of upper (lower)-type hyperfunctions 117
§4 Calculation of upper (lower)-type hyperfunctions 121
§5 Fourier transforms of upper (lower)-type hyperfunctions 124
§6 Upper (lower) power-type hyperfunctions and their Fourier trans-
forms 126
§7 Examples of application 128
§8 Summary 131
Chapter 8. FOURIER TRANSFORMS-EXISTENCE AND REGULAR-
ITY 133
§1 Preliminary 133
§2 eks a type functions and hyperfunctions 134
§3 Sufficient conditions for the existence of Fourier transforms 136
§4 Regularity of G(O = F{q'>(z)1+(z)} on the ~-axis 137
§5 Examples of application of the theorems 141
§6 Summary 143
Chapter 9. FOURIER TRANSFORM-ASYMPTOTIC BEHAVIOUR147
§ 1 Preliminary 147
§2 Riemann-Lebesgue theorem 147
§3 Reduction of hyperfunctions to ordinary functions 148
§4 Asymptotic behaviour of F{q'>(z)1+(z)} 149
