An abstract approach to approximation in spaces of pseudocontinuable functions

Abstract

We provide an abstract approach to approximation with a wide range of regularity classes X X in spaces of pseudocontinuable functions K ϑ p K^p_\vartheta , where ϑ \vartheta is an inner function and p > 0 p>0 . More precisely, we demonstrate a general principle, attributed to A. Aleksandrov, which asserts that if a certain linear manifold X X is dense in K ϑ q K^{q}_\vartheta for some q > 0 q>0 , then X X is in fact dense in K ϑ p K^p_{\vartheta } for all p > 0 p>0 . Moreover, for a rich class of Banach spaces of analytic functions X X , we describe the precise mechanism that determines when X 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.