Carleman’s Formulas in Complex Analysis

1. Front Matter

   Pages i-xx

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2. Carleman Formulas in the Theory of Functions of One Complex Variable and their Generalizations

   1. One-Dimensional Carleman Formulas

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      Pages 1-17

   2. Generalization of One-Dimensional Carleman Formulas

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      Pages 18-32

3. Carleman Formulas in Multidimensional Complex Analysis

   1. Integral Representations of Holomorphic Functions of Several Complex Variables and Logarithmic Residues

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      Pages 33-81

   2. Multidimensional Analogs of Carleman Formulas with Integration over Boundary Sets of Maximal Dimension

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      Pages 82-100

   3. Multidimensional Carleman Formulas for Sets of Smaller Dimension

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      Pages 101-128

   4. Carleman Formulas in Homogeneous Domains

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      Pages 129-142

4. First Applications

   1. Applications in Complex Analysis

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      Pages 143-162

   2. Applications in Physics and Signal Processing

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      Pages 163-191

   3. Computing Experiment

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      Pages 192-203

5. Supplement to the English Edition

   1. Criteria for Analytic Continuation. Harmonic Extension

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      Pages 204-251

   2. Carleman Formulas and Related Problems

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      Pages 252-275

6. Back Matter

   Pages 276-299

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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).

• Complex analysis
• Mathematica
• Symbol
• signal processing

• Department of Function Theory, Institute of Physics, Krasnoyarsk, Siberia

  Lev Aizenberg

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