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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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$\ell ^2$-Linear independence for the system of integer translates of a square integrable function
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by Sandra Saliani PDF
Proc. Amer. Math. Soc. 141 (2013), 937-941 Request permission

Abstract:

We prove that if the system of integer translates of a square integrable function is $\ell ^2$-linear independent, then its periodization function is strictly positive almost everywhere. Indeed we show that the above inference holds for any square integrable function since the following statement on Fourier analysis is true: For any (Lebesgue) measurable subset $A$ of $[0,1]$, with positive measure, there exists a nontrivial square summable function, with support in $A,$ whose partial sums of Fourier series are uniformly bounded in the uniform norm. This answers a question posed by Guido Weiss.
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Additional Information
  • Sandra Saliani
  • Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata, 85100 Potenza, Italia
  • Email: sandra.saliani@unibas.it
  • Received by editor(s): December 14, 2010
  • Received by editor(s) in revised form: July 20, 2011
  • Published electronically: July 17, 2012
  • Communicated by: Michael T. Lacey
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 937-941
  • MSC (2010): Primary 42C40; Secondary 42A20
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11378-4
  • MathSciNet review: 3003686