Constructive Description of Hölder Classes on Some Multidimensional Compact Sets

Abstract

In this paper, a constructive description of Holder classes of functions on certain compact sets in $${{\mathbb{R}}^{m}}$$ (m $$ \geqslant $$ 3) is given in terms of the rate of approximation by harmonic functions in shrinking neighborhoods of these compact sets. The compact sets under consideration are a higher-dimensional generalization of compact subsets of curves in $${{\mathbb{R}}^{3}}$$ whose arc is commensurable with chord. The neighborhood size is directly connected with the rate of approximation: it shrinks when the approximation is getting more accurate. In addition to being harmonic in the neighborhood of the compact, the approximating functions are subject to the condition that looks similar to the Holder condition, but is weaker. Namely, the difference in values at two points is estimated in terms of the size of the neighborhood, if the distance between these points is commensurate with the size of the neighborhood (and thus estimated in terms of the distance between the points).