An abstract approach to approximations in spaces of pseudocontinuable 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.09828
 Bibcode:
 2021arXiv210609828L
 Keywords:

 Mathematics  Functional Analysis;
 30J05;
 30J15;
 46E22;
 47B32
 EPrint:
 15 pages