Wiener's theorem for positive definite functions on hypergroups

TL;DR

This paper extends Wiener's theorem on positive definite functions from groups to a broad class of commutative hypergroups, identifying cases where the theorem holds for all exponents and characterizing functions of positive type.

Contribution

It proves Wiener's theorem for even exponents on commutative hypergroups and provides examples where the theorem holds for all exponents, unlike the classical group case.

Findings

Wiener's theorem is extended to certain hypergroups for even exponents.

Examples are given where Wiener's theorem holds for all p in [1, ∞].

Characterization of positive type functions in terms of amalgam spaces.

Abstract

The following theorem on the circle group $\mathbb{T}$ is due to Norbert Wiener: If $f\in L^{1}\left( \mathbb{T}\right) $ has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then $f\in L^{2}\left( \mathbb{T}\right) $. This result has been extended to even exponents including $p=\infty$, but shown to fail for all other $p\in\left( 1,\infty\right] .$ All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents $p\in\left[ 1,\infty\right] $. For these hypergroups and the Bessel-Kingman hypergroup with parameter $\frac{1}{2}$ we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.