Lip alpha harmonic approximation on closed sets

In this paper the Lip alpha harmonic approximation (0 < <alpha> < 1/2) on r elatively 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 Lip<alpha> harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of un iform approximation with continuous extension to the boundary for harmonic functions and similar questions in Lip alpha harmonic approximation are als o studied.