Mathematics and Computational Sciences

Some mean ergodic theorems on locally compact hypergroups

Document Type : Original Article

Authors

Seyyed Mohammad Tabatabaie
Alireza Bagheri Salec
Rasha Fahame

Department of Mathematics, University of Qom

https://doi.org/10.30511/mcs.2026.2074690.1498
Abstract
In this paper, we study the mean ergodic theorems and the weighted ones on locally compact hypergroups. Among other obtained results, for the class of all commutative hypergroups $\mathcal{H}$ with a Plancherel measure $\widetilde{\omega}$ that ${\rm supp}(\widetilde{\omega})=\widehat{\mathcal{H}}$, we prove that
if $\left(k_{j}\right)_{j \in \mathbb{N}}$ is a subsequence of $\mathbb{N}$, $f\in L^2(\mathcal{H})$, and $\mu$ is a power bounded measure on $\mathcal{H}$ such that the sequence
$$\left(\frac{1}{m} \sum_{n=1}^{m}\underbrace{\mu\ast\ldots\ast\mu}_{k_{n}-\text{times}}\ast f\right)_{m\in\mathbb{N}}$$
weakly converges in $L^2(\mathcal{H})$, then the numerical sequence $\left(\frac{1}{m} \sum_{n=1}^{m} \alpha^{k_n}\right)_{m\in\mathbb{N}}$ is convergent too for all $\alpha\in \mathbb{C}$ with
$\tilde{\omega}\left(\{\xi\in\widehat{\mathcal{H}}:\hat{\mu}(\xi)=\alpha\}\right)>0$.

Keywords

locally compact hypergroups
mean ergodic theorem
power bounded measures
${\rm w}^*$-convergence
Fourier transforms

Subjects

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Mathematics and Computational Sciences
Volume 7, Issue 1
Winter 2026
Pages 21-33

  • Receive Date 14 October 2025
  • Revise Date 27 December 2025
  • Accept Date 01 January 2026

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