Approximation of Hölder Functions in the L^p Norm by Harmonic Functions on Some Multidimensional Compact Sets

Abstract

In this paper, the class of Hölder functions in the sense of the L^p norm on certain compacts in ${{\mathbb{R}}^{m}}$ (m $\geqslant$ 3) is analyzed, and theorems of the approximation by functions being harmonic in the neighborhoods of these compacts are proved. These compacts represent a generalization of the concept of a chord-arc curve in ${{\mathbb{R}}^{3}}$ to higher dimensions. The neighborhood size decreases along with an increase in the approximation accuracy. Estimates of the approximation rate as well as the gradient of the approximation functions are made in the same L^p norm.