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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Completeness of shifted dilates in invariant Banach spaces of tempered distributions
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by Hans G. Feichtinger and Anupam Gumber PDF
Proc. Amer. Math. Soc. 149 (2021), 5195-5210 Request permission


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 $\big ( {{{\boldsymbol {L}}^2}(\mathbb {R})}, \, \|\,\cdot \,\|_{2} \big )$, 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 ${{{\boldsymbol {\mathcal {S}}}’}(\mathbb {R}^d)}$ ($d \geq 1$) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes $\big ( {{{\boldsymbol {Q}}_s}({{\mathbb {R}^{d}}})}, \, \|\,\cdot \,\|_{{{\boldsymbol {Q}}_s}} \big )$, showing that they are special cases of Katsnelson’s setting (only) for $s \geq 0$.
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Additional Information
  • Hans G. Feichtinger
  • Affiliation: NuHAG, Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
  • MR Author ID: 65680
  • ORCID: 0000-0002-9927-0742
  • Email:
  • Anupam Gumber
  • Affiliation: Department of Mathematics, Indian Institute of Science, 560012 Bangalore, India
  • MR Author ID: 1241854
  • ORCID: 0000-0001-5146-3134
  • Email:
  • Received by editor(s): August 13, 2020
  • Received by editor(s) in revised form: March 6, 2021
  • Published electronically: September 21, 2021
  • Additional Notes: The second author was supported by an Ernst Mach Grant-Worldwide Fellowship (ICM-2019-13302) from the OeAD-GmbH, Austria and NBHM-DAE (0204/19/2019R&D-II/10472)
  • Communicated by: Dmitriy Bilyk
  • © Copyright 2021 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 5195-5210
  • MSC (2020): Primary 43A15, 41A30, 43A10, 41A65, 46F05, 46B50; Secondary 43A25, 46H25, 46A40
  • DOI:
  • MathSciNet review: 4327425