-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

MathSciNet review:
3003686

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Abstract: We prove that if the system of integer translates of a square integrable function is -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 of , with positive measure, there exists a nontrivial square summable function, with support in whose partial sums of Fourier series are uniformly bounded in the uniform norm. This answers a question posed by Guido Weiss.

**1.**Eugenio Hernández, Hrvoje Šikić, Guido Weiss, and Edward Wilson,*On the properties of the integer translates of a square integrable function*, Harmonic analysis and partial differential equations, Contemp. Math., vol. 505, Amer. Math. Soc., Providence, RI, 2010, pp. 233–249. MR**2664571**, 10.1090/conm/505/09926**2.**S. V. Kisliakov,*A sharp correction theorem*, Studia Math.**113**(1995), no. 2, 177–196. MR**1318423****3.**Morten Nielsen and Hrvoje Šikić,*Schauder bases of integer translates*, Appl. Comput. Harmon. Anal.**23**(2007), no. 2, 259–262. MR**2344615**, 10.1016/j.acha.2007.04.002**4.**Maciej Paluszyński,*A note on integer translates of a square integrable function on ℝ*, Colloq. Math.**118**(2010), no. 2, 593–597. MR**2602169**, 10.4064/cm118-2-15**5.**H. Šikić and D. Speegle,*Dyadic PFW's and -bases*. In: G. Muić (ed.), Functional analysis IX, Univ. Aarhus, Aarhus (2007), 85-90.**6.**S. A. Vinogradov,*A strengthening of Kolmogorov’s theorem on the conjugate function and interpolational properties of uniformly converging power series*, Trudy Mat. Inst. Steklov.**155**(1981), 7–40, 183 (Russian). Spectral theory of functions and operators, II. MR**615564**

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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

DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11378-4

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.