Failure of Wiener's property for positive definite periodic functions

Aline Bonami; Szilárd Gy. Révész

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arXiv · 0711.0676

Failure of Wiener's property for positive definite periodic functions

Aline Bonami·Szilárd Gy. Révész

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Abstract

We say that Wiener's property holds for the exponent $p>0$ if we have that whenever a positive definite function $f$ belongs to $L^p(-ε,ε)$ for some $ε>0$, then $f$ necessarily belongs to $L^p(\TT)$, too. This holds true for $p\in 2\NN$ by a classical result of Wiener. Recently various concentration results were proved for idempotents and positive definite functions on measurable sets on the torus. These new results enable us to prove a sharp version of the failure of Wiener's property for $p\notin 2\NN$. Thus we obtain strong extensions of results of Wainger and Shapiro, who proved the negative answer to Wiener's problem for $p\notin 2\NN$.

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BibTeX RIS

Aline Bonami, Szilárd Gy. Révész. 2007-11-05. Failure of Wiener's property for positive definite periodic functions. https://arxiv.org/abs/0711.0676

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