The Spherical Transform Associated with the Generalized Gelfand Pair (U(p,q),Hn), p+q=n
Silvina Campos, Linda V. Saal · Journal of Lie Theory · 2014
We denote by H n the 2n + 1 -dimensional Heisenberg group.In this work, we study the spherical transform associated with the generalized Gelfand pair (U (p, q) H n , U (p, q)) , p + q = n , which is defined on the space of Schwartz functions on H n , and we characterize its image.In order to do that, since the spectrum associated to this pair can be identified with a subset Σ of the plane, we introduce a space H n of functions defined on R 2 and we prove that a function defined on Σ lies in the image if and only if it can be extended to a function in H n .In particular, the spherical transform of a Schwartz function f on H n admits a Schwartz extension on the plane if and only if its restriction to the vertical axis lies in S(R) .