On $C^m$-reflection of harmonic functions and $C^m$-approximation by harmonic polynomials

P. V. Paramonov ; K. Yu. Fedorovskiy

Sbornik. Mathematics, Tome 211 (2020) no. 8, pp. 1159-1170

Résumé

We obtain several new sharp $C^m$-continuity conditions, both necessary and sufficient, for operators of harmonic reflection of functions over boundaries of simple Carathéodory domains in $\mathbb R^N$. These results are based on a new criterion (also obtained in this paper) for $C^m$-continuity of the Poisson operator in the aforesaid domains. As corollaries, we give new sufficient conditions for $C^m$-approximability of functions by harmonic polynomials on boundaries of simple Carathéodory domains in $\mathbb R^N$. Bibliography: 17 titles.

Détail

Keywords: Poisson operator, harmonic reflection operator, $C^m$-approximation by harmonic polynomials.
Mots-clés : simple Carathéodory domain, Lipschitz-Hölder spaces

Bibliographie

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