The spherical transform associated with the generalized Gelfand pair \((U(p,q),H_{n}), p+q=n\) (Q2927931)

The authors consider the Heisenberg group \(H_n\) of dimension \(2n+1\) and characterize the image of the spherical transform \(F\) associated to the generalized Gelfand pair (\((U(p,q),H_{n}), p+q=n\) (Theorem 1.9). They also show that for a function \(f\) in the Schwartz class of \(H_n\), its spherical transform \(F(f)\) admits a Schwartz extension on \(\mathbb R^2\) iff its restriction to the second variable is a Schwartz function on \(\mathbb R\) (Theorem 1.11).