Harmonic Analysis
Failure of Wiener's property for positive definite periodic functions
Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 39-44.

We say that Wiener's property holds for the exponent p>0 whenever a positive definite function f, which belongs to Lp(ε,ε) for some ε>0, necessarily belongs to Lp(T), too. This holds true for p2N by a classical result of Wiener. Recently various concentration results were proved for idempotents and positive definite functions on measurable sets on the torus. They enable us to prove a sharp version of the failure of Wiener's property for p2N, strengthening results of Wainger and Shapiro.

On dit que l'exposant p possède la propriété de Wiener si toute fonction périodique définie-positive qui est de puissance p-ième intégrable au voisinage de 0 l'est sur un intervalle de période. C'est le cas des entiers pairs, d'après un résultat classique de Wiener. Nous avons récemment obtenu des phénomènes de concentration des polynômes idempotents ou définis-positifs sur un ensemble mesurable du tore qui nous permettent de donner une version forte du fait que les exposants p2N n'ont pas la propriété de Wiener, améliorant ainsi les résultats de Wainger et Shapiro.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2007.11.013
Bonami, Aline 1; Révész, Szilárd Gy. 2

1 Fédération Denis-Poisson, MAPMO-UMR 6628, département de mathématiques, Université d'Orléans, 45067 Orléans cedex 2, France
2 Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364 Hungary
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Bonami, Aline; Révész, Szilárd Gy. Failure of Wiener's property for positive definite periodic functions. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 39-44. doi : 10.1016/j.crma.2007.11.013. http://www.numdam.org/articles/10.1016/j.crma.2007.11.013/

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