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 Bochner-Schwartz theorem on locally compact abelian groups Bull. Korean Math. Soc. 2001 Vol. 38, No. 1, 7-16 Jinman Kim and Jonggyu Cho Seoul National University, Seoul National University Abstract : We study the Fourier transformation on the Gelfand--Bruhat space of type $S$ and characterize this space by means of Fourier transform on a locally compact abelian group $G$. Also, we extend Bochner--Schwartz theorem to the dual space of the Gelfand--Bruhat space and the space of Fourier hyperfunctions on $G$, respectively. Keywords : Bochner--Schwartz theorem, locally compact abelian group, positive definite, $\{M\}$-tempered MSC numbers : Primary 46F05; Secondary 43A25 Downloads: Full-text PDF