Abstract
In this paper we complete the description of Mergelyan sets for the harmonic Hardy space h ∞(B) of bounded harmonic functions in the unit ball B of , initiated in [10]. We also characterize those relatively closed subsets X of a bounded open set ω in the complex plane such that any harmonic function f on ω can be approximated uniformly on compact subsets of ω by harmonic polynomials and, simultaneously, the same sequence of polynomials converges to f uniformly on x or in Lipschitz norm on X whenever, respectively, the restriction of f to X is uniformly continuous, or is in lip(αX), 0 > α > 1/2.
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