Table Of ContentCarleman's Formulas in Complex Analysis
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
CenJre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 244
Carleman' s Formulas
in Complex Analysis
Theory and Applications
by
Lev Aizenberg
Department of Function Theory,
Institute of Physics,
Krasnoyarsk, Siberia
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Alzenberg. Lev Abramovlch. 1937-
[Formuly Karlemana v kompleksnom anallze. Engllshl
Carleman s formulas ln complex analysls I by Lev Alzenberg.
p. cm. -- (Mathematlcs and ltS appl lcatlons . v. 244)
Inc 1u des b 1 b Il ograph 1 ca 1 references and 1 ndexes.
ISBN 978-94-010-4695-4 ISBN 978-94-011-1596-4 (eBook)
DOI 10.1007/978-94-011-1596-4
1. Carleman theorem. 2. Functlons of complex varlables.
3. Mathematlcal analysls. I. Tltle. II. Serles Mathematlcs and
ltS appllcatlons (Kluwer AcademlC Publ,shersl . v. 244.
QA331.A463513 1993
515--dc20 92-43813
ISBN 978-94-010-4695-4
Printed on acid-free paper
This is an updated, enlarged and revised translation of L.A. Aizenberg,
Carleman 's Formu/as in Complex Analysis. First Applications.
Novosibirsk, Nauka Siberian Branch © 1990
AlI Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
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.
CONTENTS
Preface IX
Foreword to English Translation Xlll
Preliminaries xv
Part I. Carle man Formulas in the Theory of Functions of
One Complex Variable and their Generalizations 1
Chapter 1. One-Dimensional Carleman Formulas 1
1. Goluzin-Krylov Method 1
2. M.M. Lavrentyev's Method 9
3. Kytmanov's Method 10
4. Boundary Values of Carleman-Goluzin-Krylov Integrals 12
5. Case of the Half-Plane 14
Chapter II. Generalization of One-Dimensional Carleman
Formulas 18
6. Formula of Logarithmic Residue in the Spirit of Carleman 18
7. Carleman Formulas for the Functions of Matrices or the
Elements of a Banach Algebra 19
8. Abstract Carleman Formula 22
9. General Videnskii-Gavurina-Khavin Approach 29
Part II. Carleman Formulas in Multidimensional Complex
Analysis 33
Chapter III. Integral Representations of Holomorphic Func
tions of Several Complex Variables and Logarithmic
Residues 33
10. Martinelli-Bochner Integral Representation and
Yuzhakov-Roos Formula of Logarithmic Residue 33
11. Basic Integral Formula of Leray-Koppelman and its Corol-
laries 41
12. Multiple Cauchy Formula, Bergman-Weil Formula, Integral
Representations for Strictly Pseudoconvex and n-Circular
Domains 50
13. Andreotti-Norguet Formula and its Generalizations 58
14. Bergman Kernel Function, Szeg6 Kernel and Integral
Representations with a Holomorphic Kernel over the
Shilov Boundary 67
15. Integral Representations for Functions Holomorphic in the
Classical Domains 76
vi CONTENTS
Chapter IV. Multidimensional Analog of Carleman
Formulas with Integration over Boundary Sets of
Maximal Dimension 82
16. Carleman Formula on the basis of Martinelli-Bochner or
Cauchy-Fantappie Kernels 82
17. Theorem of Existence 90
18. Multidimensional Logarithmic Residue Formula in the
Spirit of Carleman 96
19. Carleman Formula on the Basis of the Andreotti-Norguet
Kernel 97
Chapter V. Multidimensional Carle man Formulas for
Sets of Smaller Dimension 101
20. Simplest Approaches 101
21. Carleman Formulas with Integration over One-Dimensional
Sets 110
22. Boundary Uniqueness Sets for Pluriharmonic Functions
and Reconstruction of these FUnctions 114
23. Existence of Carleman Formulas for Subsets of the Shilov
Boundary 118
Chapter VI. Carleman Formulas in Homogeneous Domains 129
24. Carleman Formulas in the Classical Domains 129
25. The Case of the Ball and Polydisk 136
26. Carleman Formulas for Siegel Domains 138
Part III. First Applications 143
Chapter VII. Applications in Complex Analysis 143
27. Criteria for Analytic Continuation into a Domain of
FUnctions Given on Part of the Boundary 143
28. Analytic Continuation from the "Edge of the Wedge" 161
Chapter VIII. Applications in Physics and Signal Processing 163
29. Examples of the Application of Carleman Formulas in
Theoretical and Mathematical Physics 163
30. Extrapolation of FUnctions Holomorphic in a Product of
Half-Planes or Strips. Analytic Continuation of the Spectrum 166
31. Interpolation of FUnctions of the Wiener Class. Analog of
the Kotelnikov Theorem for Irregular Reference Points 177
Chapter IX. Computing Experiment 192
32. Analytic Continuation of the Fourier Spectra of One-
Dimensional Finite Signals. Superresolution 192
33. Interpolation of Signals with Finite Fourier Spectrum 199
CONTENTS vii
Part IV. Supplement to the English Edition 204
Chapter X. Criteria for Analytic Continuation. Harmonic
Extension 204
34. On the Possibility of Analytic Continuation of a Function
of One Variable, Given on a Connected Boundary Arc 204
35. Some Conditions for the Harmonic Extension of Functions
in (Cm 208
36. On the Possibility of Analytic Continuation to the Domain
en
of a Function Prescribed on a Connected Part of the
Boundary 218
37. Zin's Method and its Generalizations 231
38. Some Ideas and Methods of Sections 35, 36 Applied to
Similar Problems for Harmonic Functions 245
Chapter XI. Carleman Formulas and Related Problems 252
39. New Carleman Formulas 252
40. Uniqueness in Carleman Formulas with Holomorphic Kernel 262
41. Other Results 264
Bibliography 276
~otes 288
Index of Proper ~ames 293
Subject Index 298
Index of Symbols 300
PREFACE
Integral representations of holomorphic functions play an important part in the
classical theory of functions of one complex variable and in multidimensional com
plex analysis (in the later case, alongside with integration over the whole boundary
aD of a domain D we frequently encounter integration over the Shilov boundary
5 = S(D)). They solve the classical problem of recovering at the points of a do
main D a holomorphic function that is sufficiently well-behaved when approaching
the boundary aD, from its values on aD or on S. Alongside with this classical
problem, it is possible and natural to consider the following one: to recover the
holomorphic function in D from its values on some set MeaD not containing S.
Of course, M is to be a set of uniqueness for the class of holomorphic functions
under consideration (for example, for the functions continuous in D or belonging
to the Hardy class HP(D), p ~ 1).
The first result in the direction of solving such a problem was obtained by T. Car
leman in 1926 [199] for a domain D C ([1 of special form. His idea to introduce
the "quenching" function into the integral Cauchy formula was developed in 1933
by G. M. Goluzin and V. I. Krylov [68] and applied to simply-connected plane do
mains. Their method allowed one to construct an auxiliary function, depending on
a set M, for simply-connected domains D C ([1. Generally speaking, their method
is not applicable to multiply-connected domains in ([1 or domains in ([n, n > 1.
Another method, proposed by M. M. Lavrentyev in 1956 [107] is based on approx
imating the kernel of the integral representation. This method was found to work
well in the indicated cases, also when the Goluzin-Krylov method is not applicable.
For many years the author worked on integral representations of holomorphic
functions of several complex variables. His interest in constructing Carleman for
mulas was stimulated by the appearance of the book by M. M. Lavrentyev, V. G. Ro
manov, S. P. Shishatsky [111] on ill-posed problems of analysis and mathematical
physics. The author also evoked interest in this direction among several young
mathematicians in Krasnoyarsk.
This monograph is based on several special courses delivered by the author at
the mathematical faculty of Krasnoyarsk University and contains results obtained
mainly in the last 4 years. Most of these results have not been published yet. The
author does not rule out, that the path he has taken to present them is not the best,
but the wish to publish the book sufficiently soon in order to attract the attention
of mathematicians to this new section of complex analysis played the decisive part.
The monograph consists of three parts. Part I is devoted to Carleman formulas
in the theory of functions of one complex variable, and their generalizations, in
particular, to the case of solving elliptic systems of a rather general form. We also
present a very general approach ofVidenskii-Gavurina-Khavin. This part also gives
ix
x PREFACE
the Goluzin-Krylov and Lavrentyev methods. This last method allows one to obtain
reI
the theorem of existence for (non-simply connected) domains in of Carleman
formulas with a holomorphic kernel. It is of interest that the simplest Carleman
formula that the author knows of was obtained for n = 1 and for domains of special
form by the theoretical physicists V. A. Fok and F. M. Kuni [156], although it can
also be derived from [68].
Note should be made of the Kytmanov method, which is applicable to domains
with a sufficiently rich group of automorphisms, and in the case of homogeneous
domains in ren, n > 1, allowing one to construct Carleman formulas which are,
the ball and polydisk, simpler than those obtained by other methods. As an illus
tration,the simplest case of the Kytmanov approach is given in part I for n = l.
Part II, dealing with multidimensional complex analysis, starts with chap. 3,
which presents integral representations of holomorphic functions of several complex
variables and the formulas of logarithmic residue. This chapter is for the larger
part taken from the material of the two first chapter of [34]. An exception is sec. 15
about integral representations of functions holomorphic in the classical domains,
following [221], and some new results in sec. 13. In this part the Goluzin-Krylov
method is used for sections of a domain D by complex lines or the Lavrentyev
method is applied, as well as in some cases - other approaches.
It is common knowledge that the problem of constructing a holomorphic function
in a domain D from its values on a part of the boundary aD (on a part of the
Shilov boundary), considered in this book, is not a well-posed problem. However,
it is conditionally stable, provided we confine ourselves to the class of functions
holomorphic in D and satisfying the condition If(z)1 ~ C, zED, or some other
suitable inequality. The very existence of a Carleman formula for compact K C aD
turns out to be equivalent to conditional stability of this problem for compact K.
This result is given (not only for holomorphic functions, but also for solutions of
elliptic systems of differential equations of rather general form), but the book does
not give estimates of conditional stability (for these see, for example, [Ill, 109, 112,
38].
We note that while the first two parts of the book give all Carleman formulas from
complex analysis known to the author, when writing part III the author proceeded
from his own taste. Carleman formulas have only begun to be applied recently:
nearly all results of part III were obtained in the last 2 years. We should note
sec. 27, containing a description of the traces on MeaD of functions holomorphic
in D. It is of interest that here, too, the work of V. A. Fok and F. M. Kuni [156]
in fact contains the simplest results. Besides, sec. 28 gives a brief presentation of
applications to analytical continuation "from the edge of the wedge" (a result of
A. A. Gonchar and some of its corollaries; concerning the Bogolyubov "edge of the
wedge" theorem, its generalizations and refinements see the survey by Vladimirov
[59]. We do not touch upon a number of other applications of Carleman formulas
in complex analysis (for these see, for example, [191, 192]).
By citing examples of applications of Carleman formulas to problems of theoret
ical and mathematical physics the author hopes that their number and significance
will continue to grow.
When considering problems of extrapolation and interpolation of the Fourier
spectra of finite signals (or, in essence equivalently, the finite Fourier spectrum
PREFACE xi
signals), we use the same idea as in the Goluzin-Krylov method, but with reference
to the "interior" problem of analytic continuation (the set M is lying inside the
domain D, but not on its boundary). The same results could also be obtained by
proceeding from the interpolation ideas in the book [155].
Note should be made of chap. 9, which dwells upon a computational experiment
using some earlier mentioned methods to extrapolate or interpolate signals (mostly
one-dimensional, so far) with finite spectrum, and connected with the problem of
superresolution of physical instruments, control of narrow-band noise and so on ...
A. M. Kytmanov wrote for this book chap. 6 and made a number of remarks.
L. N. Znamenskaya, N. S. Krasikova, A. M. Kytmanov, T. N. Nikitina, N. N. Tar
khanov, Yu. V. Khurumov, A. K. Tsikh, B. A. Shaimkulov provided the author with
their unpublished works. The applied results included into the book were discussed
with useful participation of M. L. Agranovskii, K. S. Aleksandrov, G. V. Alek
seyev, V. A. Ignatchenko, M. M. Lavrentyev, V. P. Palamodov, N. N. Tarkhanov.
B. A. Kravtsov and R. F. Minenkova took part in the computational experiment.
L. N. Znamenskaya and L. V. Nonkina were very helpful in preparing the man
uscript for publication. The author considers it his pleasant duty to express his
sincere gratitude to all of them.
