Universal Radial Approximation in Spaces of Analytic Functions

TL;DR

This paper extends a recent approximation result for holomorphic functions to Hardy, Bergman, and Dirichlet spaces, showing that functions in these spaces can approximate continuous functions on certain subsets of the boundary via radial limits.

Contribution

It provides new approximation theorems in classical function spaces, generalizing Charpentier's result to Hardy, Bergman, and Dirichlet spaces.

Findings

Existence of functions in Hardy spaces with boundary radial limits approximating continuous functions on zero-measure subsets.

Analogous approximation results established for Bergman and Dirichlet spaces.

Radial limit approximation holds for functions on compact subsets of the boundary with zero arc length measure.

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 $\phi$ on $K$ and any compact subset $L$ of the unit disk, there exists an increasing sequence $(r_n)_{n\in\mathbb{N}}\subseteq[0,1)$ converging to 1 such that $|f(r_n(\zeta-z)+z)-\phi(\zeta)|\to0$ as $n\to\infty$ uniformly for $\zeta\in K$ and $z\in L$. In this paper, we give analogues of this result for the Hardy spaces $H^p(\mathbb{D}),1\leq p<\infty$. 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 $H^p(\mathbb{D})$ whose radial limits can approximate every continuous function on $K$. We give similar results for the Bergman and Dirichlet spaces.