Harmonic Analysis/Mathematical Analysis
No characterization of generators in p (1<p<2) by zero set of Fourier transform
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 645-648.

Given 1<p<2 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 p(Z);

(iii) The translates of gˆ span the whole p, 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<p<2 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 p(Z) ;

(iii) Les translatées de la transformée de Fourier de g engendrent p, 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.04.017
Lev, Nir 1; Olevskii, Alexander 1

1 School of Mathematical sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
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Lev, Nir; Olevskii, Alexander. No characterization of generators in $ {\ell }^{p}$ $ (1
                  
                

[1] Beurling, A. On a closure problem, Ark. Mat., Volume 1 (1951), pp. 301-303

[2] Herz, C.S. A note on the span of translations in Lp, Proc. Amer. Math. Soc., Volume 8 (1957), pp. 724-727

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

[4] Lev, N.; Olevskii, A. Piatetski–Shapiro phenomenon in the uniqueness problem, C. R. Acad. Sci. Paris, Ser. I, Volume 340 (2005), pp. 793-798

[5] Newman, D.J. The closure of translates in lp, Amer. J. Math., Volume 86 (1964), pp. 651-667

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

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

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