Failure of Wiener's property for positive definite periodic functions

TL;DR

This paper investigates Wiener's property for positive definite functions on the torus, demonstrating its failure for certain exponents and extending previous negative results through new concentration inequalities.

Contribution

It establishes the failure of Wiener's property for all exponents not in 2N, extending prior work with sharp results and new concentration techniques.

Findings

Wiener's property holds for p in 2N due to classical results.

The property fails for p not in 2N, with sharp counterexamples.

New concentration results enable these extensions.

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 $L^p(-\epsilon,\epsilon)$ for some $\epsilon>0$, 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 $p\notin 2\NN$.