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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ \ell^2$-Linear independence for the system of integer translates of a square integrable function


Author: Sandra Saliani
Journal: Proc. Amer. Math. Soc. 141 (2013), 937-941
MSC (2010): Primary 42C40; Secondary 42A20
Published electronically: July 17, 2012
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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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Sandra Saliani
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata, 85100 Potenza, Italia
Email: sandra.saliani@unibas.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11378-4
PII: S 0002-9939(2012)11378-4
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.