Universal radial approximation in spaces of analytic functions

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 f on K and any compact subset L of the unit disk, there exists an increasing sequence (r(n))n is an element of N subset of [0, 1) converging to 1 such that |integral(r(n)(sigma - z) + z) - phi(sigma)| -> 0 as n -> infinity uniformly for zeta is an element of K and z is an element of L (see [9]). In this paper, we give analogues of this result for the Hardy spaces H-p, 1 <= p < infinity. 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 H-p (D) whose radial limits can approximate every continuous function on K. We give similar results for the Bergman and Dirichlet spaces. (C) 2022 The Author(s). Published by Elsevier Inc.