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

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.

DOI: https://doi.org/10.4213/rm180

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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

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
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• http://mi.mathnet.ru/eng/umn180
• https://doi.org/10.4213/rm180
• http://mi.mathnet.ru/eng/umn/v54/i4/p75

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. S. Ya. Khavinson, “Duality relations in the theory of analytic capacity”, St. Petersburg Math. J., 15:1 (2004), 1–40
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
4. J. E. Brennan, “Thomson's theorem on mean square polynomial approximation”, St. Petersburg Math. J., 17:2 (2006), 217–238
5. Younsi M., “On the Analytic and Cauchy Capacities”, J. Anal. Math., 135:1 (2018), 185–202
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