| Preface | |
| Foreword to English Translation | |
| Preliminaries | |
| Carleman Formulas in the Theory of Functions of One Complex Variable and their Generalizations | p. 1 |
| One-Dimensional Carleman Formulas | p. 1 |
| Goluzin-Krylov Method | p. 1 |
| M.M. Lavrentyev's Method | p. 9 |
| Kytmanov's Method | p. 10 |
| Boundary Values of Carleman-Goluzin-Krylov Integrals | p. 12 |
| Case of the Half-Plane | p. 14 |
| Generalization of One-Dimensional Carleman Formulas | p. 18 |
| Formula of Logarithmic Residue in the Spirit of Carleman | p. 18 |
| Carleman Formulas for the Functions of Matrices or the Elements of a Banach Algebra | p. 19 |
| Abstract Carleman Formula | p. 22 |
| General Videnskii-Gavurina-Khavin Approach | p. 29 |
| Carleman Formulas in Multidimensional Complex Analysis | p. 33 |
| Integral Representations of Holomorphic Functions of Several Complex Variables and Logarithmic Residues | p. 33 |
| Martinelli-Bochner Integral Representation and Yuzhakov-Roos Formula of Logarithmic Residue | p. 33 |
| Basic Integral Formula of Leray-Koppelman and its Corollaries | p. 41 |
| Multiple Cauchy Formula, Bergman-Weil Formula, Integral Representations for Strictly Pseudoconvex and n-Circular Domains | p. 50 |
| Andreotti-Norguet Formula and its Generalizations | p. 58 |
| Bergman Kernel Function, Szego Kernel and Integral Representations with a Holomorphic Kernel over the Shilov Boundary | p. 67 |
| Integral Representations for Functions Holomorphic in the Classical Domains | p. 76 |
| Multidimensional Analog of Carleman Formulas with Integration over Boundary Sets of Maximal Dimension | p. 82 |
| Carleman Formula on the basis of Martinelli-Bochner or Cauchy-Fantappie Kernels | p. 82 |
| Theorem of Existence | p. 90 |
| Multidimensional Logarithmic Residue Formula in the Spirit of Carleman | p. 96 |
| Carleman Formula on the Basis of the Andreotti-Norguet Kernel | p. 97 |
| Multidimensional Carleman Formulas for Sets of Smaller Dimension | p. 101 |
| Simplest Approaches | p. 101 |
| Carleman Formulas with Integration over One-Dimensional Sets | p. 110 |
| Boundary Uniqueness Sets for Pluriharmonic Functions and Reconstruction of these Functions | p. 114 |
| Existence of Carleman Formulas for Subsets of the Shilov Boundary | p. 118 |
| Carleman Formulas in Homogeneous Domains | p. 129 |
| Carleman Formulas in the Classical Domains | p. 129 |
| The Case of the Ball and Polydisk | p. 136 |
| Carleman Formulas for Siegel Domains | p. 138 |
| First Applications | p. 143 |
| Applications in Complex Analysis | p. 143 |
| Criteria for Analytic Continuation into a Domain of Functions Given on Part of the Boundary | p. 143 |
| Analytic Continuation from the "Edge of the Wedge" | p. 161 |
| Applications in Physics and Signal Processing | p. 163 |
| Examples of the Application of Carleman Formulas in Theoretical and Mathematical Physics | p. 163 |
| Extrapolation of Functions Holomorphic in a Product of Half-Planes or Strips. Analytic Continuation of the Spectrum | p. 166 |
| Interpolation of Functions of the Wiener Class. Analog of the Kotelnikov Theorem for Irregular Reference Points | p. 177 |
| Computing Experiment | p. 192 |
| Analytic Continuation of the Fourier Spectra of One-Dimensional Finite Signals. Superresolution | p. 192 |
| Interpolation of Signals with Finite Fourier Spectrum | p. 199 |
| Supplement to the English Edition | p. 204 |
| Criteria for Analytic Continuation. Harmonic Extension | p. 204 |
| On the Possibility of Analytic Continuation of a Function of One Variable, Given on a Connected Boundary Arc | p. 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 Boundary | p. 218 |
| Zin's Method and its Generalizations | p. 231 |
| Some Ideas and Methods of Sections 35, 36 Applied to Similar Problems for Harmonic Functions | p. 245 |
| Carleman Formulas and Related Problems | p. 252 |
| New Carleman Formulas | p. 252 |
| Uniqueness in Carleman Formulas with Holomorphic Kernel | p. 262 |
| Other Results | p. 264 |
| Bibliography | p. 276 |
| Notes | p. 288 |
| Index of Proper Names | p. 293 |
| Subject Index | p. 298 |
| Index of Symbols | p. 300 |
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