Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2022; 59(2): 367-377

Online first article February 11, 2022      Printed March 1, 2022

Copyright © The Korean Mathematical Society.

Constructions of Segal algebras in $L^1(G)$ of LCA groups $G$ in which a generalized Poisson summation formula holds

Jyunji Inoue, Sin-Ei Takahasi

Hokkaido University; Laboratory of Mathematics and Games


Let $G$ be a non-discrete locally compact abelian group, and $\mu$ be a transformable and translation bounded Radon measure on $G$. In this paper, we construct a Segal algebra $S_{\mu}(G)$ in $L^1(G)$ such that the generalized Poisson summation formula for $\mu$ holds for all $f\in S_{\mu}(G)$, for all $x\in G$. For the definitions of transformable and translation bounded Radon measures and the generalized Poisson summation formula, we refer to L. Argabright and J. Gil de Lamadrid's monograph in 1974.

Keywords: Locally compact abelian group, group algebra, Segal algebra, Radon measure, transformable measure, translation bounded measure, shift-bounded measure, Fourier transform, Poisson summation formula, generalized Poisson summation formula

MSC numbers: Primary 43A20; Secondary 42A38, 43A25