Universal radial approximation in spaces of analytic functions

Abstract

Recently, Charpentier showed that there exist holomorphic functions f in the unit disk such that, for any proper compact subset K of the unit circle, any continuous function ϕ on K and any compact subset L of the unit disk, there exists an increasing sequence (rn)n∈N⊆[0,1) converging to 1 such that |f(rn(ζ−z)+z)−ϕ(ζ)|→0 as n→∞ uniformly for ζ∈K and z∈L (see [9]). In this paper, we give analogues of this result for the Hardy spaces Hp,1≤p<∞. In particular, our main result implies that, if we fix a compact subset K of the unit circle with zero arc length measure, then there exist functions in Hp whose radial limits can approximate every continuous function on K. We give similar results for the Bergman and Dirichlet spaces.