Spherical transforms and Plancherel formulae

2007 ◽  
pp. 179-205
Author(s):  
Joseph Wolf
Keyword(s):  
Spherical Transforms

A Paley-Wiener Theorem for the Spherical Transform Associated with the Generalized Gelfand Pair $(U(p,q),H_{n})$, $p+q=n$

2016 ◽  
Vol 119 (2) ◽  
pp. 249
Author(s):  
Silvina Campos
Keyword(s):  
Heisenberg Group ◽  
Holomorphic Functions ◽  
Real Variable ◽  
Gelfand Pair ◽  
Wiener Theorem ◽  
Spherical Transforms

In this work we prove a Paley-Wiener theorem for the spherical transform associated to the generalized Gelfand pair $(H_n\ltimes U(p,q),H_n)$, where $H_n$ is the $2n+1$-dimensional Heisenberg group. In particular, by using the identification of the spectrum of $(U(p,q),H_n)$ with a subset $\Sigma$ of $\mathbb{R}^2$, we prove that the restrictions of the spherical transforms of functions in $C_{0}^{\infty}(H_n)$ to appropriated subsets of $\Sigma$, can be extended to holomorphic functions on $\mathbb{C}^2$. Also, we obtain a real variable characterizations of such transforms.

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