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Uspekhi Mat. Nauk, 1999, Volume 54, Issue 4(328), Pages 75–142 (Mi umn180)  

This article is cited in 4 scientific papers (total in 5 papers)

Golubev sums: a theory of extremal problems like the analytic capacity problem and of related approximation processes

S. Ya. Havinson

Moscow State University of Civil Engineering

Abstract: We study analogues of analytic capacity for classes of analytic functions representable via some special analytic machinery, which we refer to as “Golubev sums”. A Golubev sum contains derivatives of various (given) orders of Cauchy potentials (in particular, the Cauchy potentials themselves can occur in a Golubev sum). Furthermore, the measures determining distinct terms of a Golubev sum are in general defined on distinct compact sets. We consider Golubev sums with various types of measures: complex, real, and positive. We present an abstract scheme for studying extremal problems like the analytic capacity problem. The dual problems turn out to be approximation problems in which the size of the approximants is taken into account. In the case of positive measures, the approximation problem is transformed into a problem in which one has to move a given element of a space into a given cone in that space by adding linear combinations of elements of a given subspace with coefficients as small as possible. As a preliminary, we state criteria for the representability of an analytic function by Golubev sums of various kinds. These criteria generalize known criteria for representability by Cauchy potentials.


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English version:
Russian Mathematical Surveys, 1999, 54:4, 753–818

Bibliographic databases:

UDC: 517.535.4
MSC: Primary 30C85; Secondary 31A15, 30D10, 30C70
Received: 04.03.1998

Citation: S. Ya. Havinson, “Golubev sums: a theory of extremal problems like the analytic capacity problem and of related approximation processes”, Uspekhi Mat. Nauk, 54:4(328) (1999), 75–142; Russian Math. Surveys, 54:4 (1999), 753–818

Citation in format AMSBIB
\by S.~Ya.~Havinson
\paper Golubev sums: a theory of extremal problems like the analytic capacity problem and of related approximation processes
\jour Uspekhi Mat. Nauk
\yr 1999
\vol 54
\issue 4(328)
\pages 75--142
\jour Russian Math. Surveys
\yr 1999
\vol 54
\issue 4
\pages 753--818

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    This publication is cited in the following articles:
    1. S. Ya. Havinson, “Approximations by wedge elements taking into account the values of the approximating elements”, Russian Math. (Iz. VUZ), 46:10 (2002), 69–82  mathnet  mathscinet  zmath  elib
    2. S. Ya. Khavinson, “Duality relations in the theory of analytic capacity”, St. Petersburg Math. J., 15:1 (2004), 1–40  mathnet  crossref  mathscinet  zmath
    3. A. G. Vitushkin, A. A. Gonchar, M. V. Samokhin, V. M. Tikhomirov, P. L. Ul'yanov, V. P. Havin, V. Ya. Èiderman, “Semën Yakovlevich Khavinson (obituary)”, Russian Math. Surveys, 59:4 (2004), 777–785  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. J. E. Brennan, “Thomson's theorem on mean square polynomial approximation”, St. Petersburg Math. J., 17:2 (2006), 217–238  mathnet  crossref  mathscinet  zmath  elib
    5. Younsi M., “On the Analytic and Cauchy Capacities”, J. Anal. Math., 135:1 (2018), 185–202  crossref  mathscinet  zmath  isi  scopus
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