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

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 $R^N$. 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 $R^N$.

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