Wiener's theorem for positive definite functions on hypergroups

• Published in: arXiv.org 2014-05
• Main Authors: Bloom, Walter R, Fournier, John J F, Leinert, Michael
• Format: Article
• Language: English
• Subjects: Exponents; Group theory; Mathematical functions; Theorems
• ISSN: 2331-8422

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.