Wiener's theorem on hypergroups
| Autor: | Michael Leinert, John J. F. Fournier, Walter R. Bloom |
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| Rok vydání: | 2020 |
| Předmět: |
Discrete mathematics
Control and Optimization Algebra and Number Theory Integrable system positive definite amalgam space Positive-definite matrix Bessel-Kingman Circle group 510 Mathematics Square-integrable function Wiener 43A62 (strong) hypergroup 43A35 Locally compact space Abelian group Commutative property Fourier series Analysis 43A15 Mathematics |
| Zdroj: | Ann. Funct. Anal. 6, no. 4 (2015), 30-59 |
| DOI: | 10.11588/heidok.00029228 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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