Article
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Representations by Beurling Systems
Version 1
: Received: 7 August 2023 / Approved: 8 August 2023 / Online: 8 August 2023 (13:00:18 CEST)
A peer-reviewed article of this Preprint also exists.
Kazarian, K. Representations by Beurling Systems. Mathematics 2023, 11, 3663. Kazarian, K. Representations by Beurling Systems. Mathematics 2023, 11, 3663.
Abstract
We say that a system {zmF(z)}m=0∞ is a Beurling system if F is an outer function. Beurling’s approximation theorem asserts that if F is an outer function from H2(D) then the system {zmF(z)}m=0∞ is complete in the space H2(D). We prove that a Beurling system with F∈Hp(D),1≤p<∞ is an M−bases in Hp(D) with an explicit dual system. Any function f∈Hp(D),1≤p<∞ can be expanded as a series by the system {zmF(z)}m=0∞. For different methods of summation we characterize outer functions F for which the expansion converges to f. Related results for weighted Hardy spaces in the unit disc are studied. Particularly we prove Rosenblum’s hypothesis.
Keywords
summation basis; Hardy spaces; outer function; Beurling system; kernels; representation of functions
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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