$\ell ^2$-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.References
- 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, DOI 10.1090/conm/505/09926
- S. V. Kisliakov, A sharp correction theorem, Studia Math. 113 (1995), no. 2, 177–196. MR 1318423, DOI 10.4064/sm-113-2-177-196
- Morten Nielsen and Hrvoje Šikić, Schauder bases of integer translates, Appl. Comput. Harmon. Anal. 23 (2007), no. 2, 259–262. MR 2344615, DOI 10.1016/j.acha.2007.04.002
- Maciej Paluszyński, A note on integer translates of a square integrable function on $\Bbb R$, Colloq. Math. 118 (2010), no. 2, 593–597. MR 2602169, DOI 10.4064/cm118-2-15
- H. Šikić and D. Speegle, Dyadic PFW’s and $W_o$-bases. In: G. Muić (ed.), Functional analysis IX, Univ. Aarhus, Aarhus (2007), 85–90.
- 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
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