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Completeness of shifted dilates in invariant Banach spaces of tempered distributions

Feichtinger, HG and Gumber, A (2021) Completeness of shifted dilates in invariant Banach spaces of tempered distributions. In: Proceedings of the American Mathematical Society, 149 (12). pp. 5195-5210.

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Official URL: https://doi.org/10.1090/proc/15564


We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V. Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into (L2(ℝ), || · ||2), nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space S'(ℝd) (d ≥ 1) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes (Qs(ℝd), || · ||Qs), showing that they are special cases of Katsnelson’s setting (only) for s ≥ 0. ©2021 American Mathematical Society

Item Type: Journal Article
Publication: Proceedings of the American Mathematical Society
Publisher: American Mathematical Society
Additional Information: The copyright for this article belongs to the Authors.
Keywords: Approximation by translations; Banach modules; Banach spaces of tempered distributions; Beurling algebra; Compactness; Modulation spaces; Shubin spaces
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 06 Jun 2023 10:17
Last Modified: 06 Jun 2023 10:17
URI: https://eprints.iisc.ac.in/id/eprint/81823

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