We obtain several new sharp ##IMG## [http://ej.iop.org/images/1064-5616/211/8/1159/toc_MSB_211_8_1159ieqn1.gif] {$C^m$} -continuity conditions, both necessary and sufficient, for operators of harmonic reflection of functions over boundaries of simple Carathéodory domains in ##IMG## [http://ej.iop.org/images/1064-5616/211/8/1159/MSB_211_8_1159ieqn2.gif] {$\mathbb R^N$} . These results are based on a new criterion (also obtained in this paper) for ##IMG## [http://ej.iop.org/images/1064-5616/211/8/1159/toc_MSB_211_8_1159ieqn1.gif] {$C^m$} -continuity of the Poisson operator in the aforesaid domains. As corollaries, we give new sufficient conditions for ##IMG## [http://ej.iop.org/images/1064-5616/211/8/1159/toc_MSB_211_8_1159ieqn1.gif] {$C^m$} -approximability of functions by harmonic polynomials on boundaries of simple Carathéodory domains in ##IMG## [http://ej.io...] {$\mathbb R^N$}

中文翻译：

---

在## IMG ##上[http://ej.iop.org/images/1064-5616/211/8/1159/toc_MSB_211_8_1159ieqn1.gif] {$ C ^ m $}-谐波函数和## IMG＃的反射＃[http://ej.iop.org/images/1064-5616/211/8/1159/toc_MSB_211_8_1159ieqn1.gif] {$ C ^ m $}-通过谐波极化近似...

我们获得了几个新的锋利的## IMG ## [http://ej.iop.org/images/1064-5616/211/8/1159/toc_MSB_211_8_1159ieqn1.gif] {$ C ^ m $}-连续性条件，都必须对于在## IMG ## [http://ej.iop.org/images/1064-5616/211/8/1159/MSB_211_8_1159/MSB_211_8_1159ieqn2.gif] { $ \ mathbb R ^ N $}。这些结果基于## IMG ## [http://ej.iop.org/images/1064-5616/211/8/1159/toc_MSB_211_8_1159ieqn1.gif] {$在上述域中的泊松算子的连续性。作为推论，我们为## IMG ## [http://ej.iop.org/images/1064-5616/211/8/1159/toc_MSB_211_8_1159ieqn1提供了新的充分条件。