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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Complex Fourier analysis on a nilpotent Lie group


Author: Roe Goodman
Journal: Trans. Amer. Math. Soc. 160 (1971), 373-391
MSC: Primary 22E30; Secondary 22E45
MathSciNet review: 0417334
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Abstract: Let $ G$ be a simply-connected nilpotent Lie group, with complexification $ {G_c}$. The functions on $ G$ which are analytic vectors for the left regular representation of $ G$ on $ {L_2}(G)$ are determined in this paper, via a dual characterization in terms of their analytic continuation to $ {G_c}$, and by properties of their $ {L_2}$ Fourier transforms. The analytic continuation of these functions is shown to be given by the Fourier inversion formula. An explicit construction is given for a dense space of entire vectors for the left regular representation. In the case $ G = R$ this furnishes a group-theoretic setting for results of Paley and Wiener concerning functions holomorphic in a strip.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0417334-3
PII: S 0002-9947(1971)0417334-3
Keywords: Nilpotent Lie group, Plancherel theorem, Fourier inversion formula, Paley-Wiener theorem, analytic vector, entire vector, analytic continuation of representations
Article copyright: © Copyright 1971 American Mathematical Society