𝐿𝑖𝑝𝛼 harmonic approximation on closed sets

Abstract

In this paper the L i p α Lip\alpha harmonic approximation ( 0 > α > 1 2 0 > \alpha > \frac {1}{2} ) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on L i p α Lip\alpha harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in L i p α Lip\alpha harmonic approximation are also studied.