A Paley-Wiener Theorem for the Spherical Transform Associated with the Generalized Gelfand Pair $(U(p,q),H_{n})$, $p+q=n$ - Campos, Silvina (MATHEMATICA SCANDINAVICA, 2016)
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 $Σ$ of $\mathbb{R}^2$, we prove that the restrictions of the spherical transforms of functions in $C_{0}^{∞}(H_n)$ to appropriated subsets of $Σ$, can be extended to holomorphic functions on $\mathbb{C}^2$. Also, we obtain a real variable characterizations of such transforms.