On the Schwartz-Bruhat space and the Paley-Wiener theorem for locally compact abelian groups

AbstractLet G be a locally compact abelian group. The Schwartz-Bruhat space of functions on G is then defined in terms of Lie subquotient groups. We give an alternative characterization which involves asymptotic behavior of the function and its Fourier transform, and which makes no reference to Lie theory. We then prove the Paley-Wiener theorem for the Fourier transform of CC∞(G). The asymptotic estimates which arise are closely related to those used to characterize the Schwartz-Bruhat space

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