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Carleman's Formulas in Complex Analysis : MATHEMATICS AND ITS APPLICATIONS (KLUWER ) - L. A. Aizenberg

Carleman's Formulas in Complex Analysis

MATHEMATICS AND ITS APPLICATIONS (KLUWER )

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This monograph is the first to give a systematic presentation of the Carleman formulas. These enable values of functions holomorphic to a domain to be recovered from their values over a part of the boundary of the domain. Various generalizations of these formulas are considered. Applications are considered to problems of analytic continuation in the theory of functions, and, in a broader context, to problems arising in theoretical and mathematical physics, and to the extrapolation and interpolation of signals having a finite Fourier spectrum. The volume also contains a review of the latest results, including those obtained by computer simulation on the elimination of noise in a given frequency band. For mathematicians and theoretical physicists whose work involves complex analysis, and those interested in signal processing.

Preface
Foreword to English Translation
Preliminaries
Carleman Formulas in the Theory of Functions of One Complex Variable and their Generalizationsp. 1
One-Dimensional Carleman Formulasp. 1
Goluzin-Krylov Methodp. 1
M.M. Lavrentyev's Methodp. 9
Kytmanov's Methodp. 10
Boundary Values of Carleman-Goluzin-Krylov Integralsp. 12
Case of the Half-Planep. 14
Generalization of One-Dimensional Carleman Formulasp. 18
Formula of Logarithmic Residue in the Spirit of Carlemanp. 18
Carleman Formulas for the Functions of Matrices or the Elements of a Banach Algebrap. 19
Abstract Carleman Formulap. 22
General Videnskii-Gavurina-Khavin Approachp. 29
Carleman Formulas in Multidimensional Complex Analysisp. 33
Integral Representations of Holomorphic Functions of Several Complex Variables and Logarithmic Residuesp. 33
Martinelli-Bochner Integral Representation and Yuzhakov-Roos Formula of Logarithmic Residuep. 33
Basic Integral Formula of Leray-Koppelman and its Corollariesp. 41
Multiple Cauchy Formula, Bergman-Weil Formula, Integral Representations for Strictly Pseudoconvex and n-Circular Domainsp. 50
Andreotti-Norguet Formula and its Generalizationsp. 58
Bergman Kernel Function, Szego Kernel and Integral Representations with a Holomorphic Kernel over the Shilov Boundaryp. 67
Integral Representations for Functions Holomorphic in the Classical Domainsp. 76
Multidimensional Analog of Carleman Formulas with Integration over Boundary Sets of Maximal Dimensionp. 82
Carleman Formula on the basis of Martinelli-Bochner or Cauchy-Fantappie Kernelsp. 82
Theorem of Existencep. 90
Multidimensional Logarithmic Residue Formula in the Spirit of Carlemanp. 96
Carleman Formula on the Basis of the Andreotti-Norguet Kernelp. 97
Multidimensional Carleman Formulas for Sets of Smaller Dimensionp. 101
Simplest Approachesp. 101
Carleman Formulas with Integration over One-Dimensional Setsp. 110
Boundary Uniqueness Sets for Pluriharmonic Functions and Reconstruction of these Functionsp. 114
Existence of Carleman Formulas for Subsets of the Shilov Boundaryp. 118
Carleman Formulas in Homogeneous Domainsp. 129
Carleman Formulas in the Classical Domainsp. 129
The Case of the Ball and Polydiskp. 136
Carleman Formulas for Siegel Domainsp. 138
First Applicationsp. 143
Applications in Complex Analysisp. 143
Criteria for Analytic Continuation into a Domain of Functions Given on Part of the Boundaryp. 143
Analytic Continuation from the "Edge of the Wedge"p. 161
Applications in Physics and Signal Processingp. 163
Examples of the Application of Carleman Formulas in Theoretical and Mathematical Physicsp. 163
Extrapolation of Functions Holomorphic in a Product of Half-Planes or Strips. Analytic Continuation of the Spectrump. 166
Interpolation of Functions of the Wiener Class. Analog of the Kotelnikov Theorem for Irregular Reference Pointsp. 177
Computing Experimentp. 192
Analytic Continuation of the Fourier Spectra of One-Dimensional Finite Signals. Superresolutionp. 192
Interpolation of Signals with Finite Fourier Spectrump. 199
Supplement to the English Editionp. 204
Criteria for Analytic Continuation. Harmonic Extensionp. 204
On the Possibility of Analytic Continuation of a Function of One Variable, Given on a Connected Boundary Arcp. 204
Some Conditions for the Harmonic Extension of Functions in C[superscript n]p. 208
On the Possibility of Analytic Continuation to the Domain C[superscript n] of a Function Prescribed on a Connected Part of the Boundaryp. 218
Zin's Method and its Generalizationsp. 231
Some Ideas and Methods of Sections 35, 36 Applied to Similar Problems for Harmonic Functionsp. 245
Carleman Formulas and Related Problemsp. 252
New Carleman Formulasp. 252
Uniqueness in Carleman Formulas with Holomorphic Kernelp. 262
Other Resultsp. 264
Bibliographyp. 276
Notesp. 288
Index of Proper Namesp. 293
Subject Indexp. 298
Index of Symbolsp. 300
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792321217
ISBN-10: 0792321219
Series: MATHEMATICS AND ITS APPLICATIONS (KLUWER )
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 299
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.88  x 2.54
Weight (kg): 0.59