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).
Table of ContentsPreface. Foreword to the English Translation. Preliminaries. Part I: Carleman Formulas in the Theory of Functions of One Complex Variable and their Generalization. I. One-Dimensional Carleman Formulas. II. Generalizations of One-Dimensional Carleman Formulas. Part II: Carleman Formulas in Multidimensional Complex Analysis. III. Integral Representations of Holomorphic Functions of Several Complex Variables and Logarithmic Residues. IV. Multidimensional Analog of Carleman Formulas with Integration over the Boundary Sets of Maximal Dimension. V. Multidimensional Carleman Formulas for Sets of Smaller Dimension. VI. Carleman Formulas in Homogeneous Domains. Part III: First Applications. VII. Applications in Complex Analysis. VIII. Applications in Physics and Signal Processing. IX. Computing Experiment. Part IV: Supplement to English Edition. X. Criteria for Analytic Continuation Harmonic Extension. XI. Carleman Formulas and Related Problems. Bibliography. Notes. Index of Proper Names. Subject Index. Index of Symbols.