Harmonic Analysis/Mathematical Analysis
No characterization of generators in $ℓp$ $(1 by zero set of Fourier transform
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 645-648.

Given $1 we construct two continuous functions f and g on the circle, with the following properties:

(i) They have the same set of zeros;

(ii) The Fourier transforms $fˆ$ and $gˆ$ both belong to $ℓp(Z)$;

(iii) The translates of $gˆ$ span the whole $ℓp$, but those of $fˆ$ do not.

A similar result is true for $Lp(R)$. This should be contrasted with the Wiener theorems related to $p=1,2$.

Étant donné $1 nous construisons deux fonctions continues sur le cercle, f et g, telles que :

(i) Elles ont le même ensemble de zéros ;

(ii) Leurs transformées de Fourier appartiennent à $ℓp(Z)$ ;

(iii) Les translatées de la transformée de Fourier de g engendrent $ℓp$, mais non celles de la transformées de Fourier de f.

Un résultat analogue est valable pour $Lp(R)$. Cela contraste avec les cas $p=1$ ou 2, élucidés par Wiener.

Accepted:
Published online:
DOI: 10.1016/j.crma.2008.04.017

Nir Lev 1; Alexander Olevskii 1

1 School of Mathematical sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
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author = {Nir Lev and Alexander Olevskii},
title = {No characterization of generators in ${\ell }^{p}$ $(1<p<2)$ by zero set of {Fourier} transform},
journal = {Comptes Rendus. Math\'ematique},
pages = {645--648},
publisher = {Elsevier},
volume = {346},
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year = {2008},
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Nir Lev; Alexander Olevskii. No characterization of generators in ${\ell }^{p}$ \$ (1



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[3] J.-P. Kahane; R. Salem Ensembles parfaits et séries trigonométriques, Hermann, 1994

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[6] H. Pollard The closure of translations in $Lp$, Proc. Amer. Math. Soc., Volume 2 (1951), pp. 100-104

[7] N. Wiener The Fourier Integral and Certain of its Applications, Cambridge University Press, 1933 (Reprint, Dover Publications, 1959)

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