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Mat. Sb., 2015, Volume 206, Number 2, Pages 77–118 (Mi msb8334)  

This article is cited in 1 scientific paper (total in 1 paper)

Criteria for $C^m$-approximability by bianalytic functions on planar compact sets

M. Ya. Mazalova, P. V. Paramonovb

a National Research University "Moscow Power Engineering Institute" in Smolensk
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The paper puts forward criteria for approximability by bianalytic functions in the norms of the Whitney-type spaces $C^m$ on planar compact sets with $m \in (0, 2)$. These results, which are analogues of Vitushkin's well-known criteria for uniform rational approximation, together with results of O'Farrell and Verdera (the case $m \ge 2$) and Mazalov (the case $m=0$), provide a complete set of criteria for approximability by bianalytic functions for all $m \ge 0$. These conditions for approximability are obtained for both individual functions and (as corollaries) for classes of functions, using the terminology of geometric measure theory.
Bibliography: 21 titles.

Keywords: $C^m$-approximation by bianalytic functions, bianalytic $C^m$-capacity, Hausdorff content of order $m$, Vitushkin-type localization operator.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation НШ-2900.2014.1
Russian Foundation for Basic Research 12-01-00434-а


DOI: https://doi.org/10.4213/sm8334

Full text: PDF file (838 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2015, 206:2, 242–281

Bibliographic databases:

UDC: 517.53+517.548.4
MSC: Primary 30E10; Secondary 31A99, 31C35
Received: 02.02.2014 and 24.04.2014

Citation: M. Ya. Mazalov, P. V. Paramonov, “Criteria for $C^m$-approximability by bianalytic functions on planar compact sets”, Mat. Sb., 206:2 (2015), 77–118; Sb. Math., 206:2 (2015), 242–281

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. P. V. Paramonov, “Criteria for the individual $C^m$-approximability of functions on compact subsets of $\mathbb R^N$ by solutions of second-order homogeneous elliptic equations”, Sb. Math., 209:6 (2018), 857–870  mathnet  crossref  crossref  adsnasa  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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