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 $X$ 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. Aleksandrov, which asserts that if a certain linear manifold $X$ is dense in $K^{q}_\vartheta$ for some $q>0$, then $X$ is in fact dense in $K^p_{\vartheta }$ for all $p>0$. 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.