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Transactions of the American Mathematical Society

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A Paley-Wiener theorem for locally compact abelian groups

Author: Gunar E. Liepins
Journal: Trans. Amer. Math. Soc. 222 (1976), 193-210
MSC: Primary 43A32
MathSciNet review: 0430679
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Abstract: Extending the Paley-Wiener theorem to locally compact Abelian groups involves both finding a suitable Laplace transform and a suitable analogue for analytic functions. The Laplace transform is defined in terms of complex characters, and differentiability is defined with use of one-parameter subgroups. The resulting theorem is much as conjectured by Mackey [7],($^{1}$) the major differences being that the theorem is very much an ${L^2}$ theorem and that the problem exhibits a surprising finite dimensional nature.

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Keywords: Locally compact Abelian groups, Laplace transform, group character
Article copyright: © Copyright 1976 American Mathematical Society