Constructions of Segal algebras in $L^1(G)$ of LCA groups $G$ in which a generalized Poisson summation formula holds
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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.
Abstract
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.
참고문헌 (5)
- L. Argabright, "Fourier analysis of unbounded measures on lo-cally compact abelian groups" (1974), American Mathematical Society
- C. Berg, "Potential theory on locally compact abelian groups" (1975), Springer-Verlag
- H. Reiter, "L1-algebras and Segal algebras" (1971), Springer-Verlag
- H. Reiter, "Classical harmonic analysis and locally compact groups" (2000), The Claren-don Press, Oxford University Press
- W. Rudin, "Fourier analysis on groups" (1962), Interscience Publishers (a division of John Wiley and Sons)