An abstract approach to approximations in spaces of pseudocontinuable functions

TL;DR

This paper presents an abstract framework for approximation in spaces of pseudocontinuable functions, extending density results across various regularity classes and inner functions, with implications for boundary behavior and Taylor series convergence.

Contribution

It generalizes Aleksandrov's density theorem to broader classes of functions and provides a mechanism to determine density in spaces of pseudocontinuable functions.

Findings

Density of linear manifolds in one space implies density in all related spaces.

Extension of Aleksandrov's theorem to functions with uniformly convergent Taylor series.

Framework applicable to a wide range of regularity classes and inner functions.

Abstract

We give an abstract approach to approximations with a wide range of regularity classes $X$ in spaces of pseudocontinuable functions $K^p_\vartheta$, where $\vartheta$ is an inner function and $p>0$. More precisely, we demonstrate a general principle, attributed to A. B. Aleksandrov, which asserts that if a certain linear manifold $X$ is dense in the space of pseudocontinuable functions $K^{p_0}_\vartheta$, for some $p_0>0$, then $X$ is in fact dense in $K^p_{\vartheta}$, for all $p>0$. %This allows for generalizations of the recent result on density by functions with smooth boundary extensions. Moreover, for a rich class of Banach spaces of analytic functions $X$, we describe the precise mechanism that determines when $X$ is dense in a certain space of pseudocontinuable functions. As a consequence, we obtain an extension of Aleksandrov's density theorem to the class of analytic functions with uniformly convergent Taylor series.