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An abstract approach to approximation in spaces of pseudocontinuable functions


Authors: Adem Limani and Bartosz Malman
Journal: Proc. Amer. Math. Soc. 150 (2022), 2509-2519
MSC (2020): Primary 46E22; Secondary 30J05
DOI: https://doi.org/10.1090/proc/15864
Published electronically: March 16, 2022
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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.


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Additional Information

Adem Limani
Affiliation: Centre for Mathematical Sciences, Lund University, SE-22 100, Lund, Sweden
ORCID: 0000-0002-5938-3670
Email: adem.limani@math.lu.se; ademlimani@gmail.com

Bartosz Malman
Affiliation: Royal Institute of Technology, KTH, SE-100 44, Stockholm, Sweden
MR Author ID: 1231604
Email: malman@kth.se

Received by editor(s): August 25, 2021
Received by editor(s) in revised form: September 19, 2021, and September 21, 2021
Published electronically: March 16, 2022
Additional Notes: The first author is the corresponding author
Communicated by: Javad Mashreghi
Article copyright: © Copyright 2022 American Mathematical Society