Annals of Functional Analysis

Wiener's theorem on hypergroups

John J. F. Fournier, Michael Leinert, and Walter R. Bloom

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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.

Article information

Ann. Funct. Anal., Volume 6, Number 4 (2015), 30-59.

First available in Project Euclid: 1 July 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 43A62: Hypergroups
Secondary: 43A35: Positive definite functions on groups, semigroups, etc. 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.

Wiener (strong) hypergroup positive definite amalgam space Bessel-Kingman


R. Bloom, Walter; Fournier, John J. F.; Leinert, Michael. Wiener's theorem on hypergroups. Ann. Funct. Anal. 6 (2015), no. 4, 30--59. doi:10.15352/afa/06-4-30.

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