Wiener's theorem for positive definite functions on hypergroups

Abstract

The following theorem on the circle group T is due to Norbert Wiener: If f ∈ L 1 (T) has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then f ∈ L 2 (T). This result has been extended to even exponents including p = ∞, but shown to fail for all other p ∈ (1, ∞] . 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 ∈ [1, ∞]. For these hypergroups and the Bessel-Kingman hypergroup with parameter 1 2 we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.