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Wiener's theorem on hypergroups

Bloom, W.R., Fournier, J.J.F. and Leinert, M. (2015) Wiener's theorem on hypergroups. Annals of Functional Analysis, 6 (4). pp. 30-59.

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The following theorem on the circle group T is due to Norbert Wiener: If f∈L1(T) has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then f∈L2(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 12 we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.

Publication Type: Journal Article
Murdoch Affiliation: School of Engineering and Information Technology
Publisher: Duke University Press
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