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Mat. Sb., 2018, Volume 209, Number 6, Pages 83–97 (Mi msb8967)  

Criteria for the individual $C^m$-approximability of functions on compact subsets of $\mathbb R^N$ by solutions of second-order homogeneous elliptic equations

P. V. Paramonovab

a Faculty of Mechanics and Mathematicsб Lomonosov Moscow State University
b Saint Petersburg State University

Abstract: Criteria for the individual approximability of functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients in the norms of Whitney-type $C^m$-spaces on compact subsets of $\mathbb R^N$, $N\in\{2,3,…\}$, are obtained for $m \in (0, 1) \cup (0,2)$. These results, which are analogues of Vitushkin's celebrated criteria for uniform rational approximation, were previously established by Mazalov for harmonic approximations (for $m \in (0, 1)$ and $N \geq 3$) and by Mazalov and Paramonov for bi-analytic approximation.
Bibliography: 11 titles.

Keywords: $C^m$-approximation by solutions of homogeneous elliptic equations, Vitushkin-type localization operator, $C^m$-invariance of Calderón-Zygmund operators, $p$-dimensional Hausdorff content, harmonic $C^m$-capacity, $L$-oscillation.

Funding Agency Grant Number
Russian Science Foundation 17-11-01064
The work was supported by the Russian Science Foundation under grant no. 17-11-01064.


DOI: https://doi.org/10.4213/sm8967

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English version:
Sbornik: Mathematics, 2018, 209:6, 857–870

Bibliographic databases:

UDC: 517.518.8+517.57+517.956.22
MSC: Primary 41A30; Secondary 35J15, 42B20
Received: 16.05.2017

Citation: 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”, Mat. Sb., 209:6 (2018), 83–97; Sb. Math., 209:6 (2018), 857–870

Citation in format AMSBIB
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