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The Spherical Transform of a Schwartz Function on the Heisenberg Group

✍ Scribed by Chal Benson; Joe Jenkins; Gail Ratcliff

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 513 KB
Volume: 154
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1997.3197

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✦ Synopsis

Suppose that K/U(n) is a compact Lie group acting on the (2n+1)-dimensional Heisenberg group H n . We say that (K, H n ) is a Gelfand pair if the convolution algebra L 1 K (H n ) of integrable K-invariant functions on H n is commutative. In this case, the Gelfand space 2(K, H n ) is equipped with the Godement Plancherel measure, and the spherical transform 7 :

) is an isometry. The main result in this paper provides a complete characterization of the set

We show that a function F on 2(K, H n ) belongs to S K (H n ) 7 if and only if the functions obtained from F via application of certain derivatives and difference operators satisfy decay conditions. We also consider spherical series expansions for K-invariant Schwartz functions on H n modulo its center.

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