The authors develop a new approach to the spherical Fourier analysis on a Gelfand pair by introducing a space of test functions consisting of linear combinations of Gaussian functions. They define the space Gauss\(({\mathbb R})\) as generated by the functions \(\varphi _c(x)=e^{-cx^2}\) (\(c>0\)) and, for a function \(q\) of the form \[ q(x)=\sum _{n=0}^{\infty } a_nx^{2n}\;\text{ with }\;a_n\geq 0, \] satisfying further \(q(x)={\mathcal O}(e^{b| x| })\) for some \(b>0\) and, for all \(A>0\), \[ \sum _{n=0}^{\infty } a_nA^nn!<\infty , \] (one says that \(q\) is admissible) the space \(q\)-Gauss\(({\mathbb R})\) is generated by the functions \(q\varphi _c\).
The authors prove that \(q\)-Gauss\(({\mathbb R})\) is contained in the closure of the space Gauss\(({\mathbb R})\), and further that it holds also for \({\mathcal C}_c({\mathbb R})\). Then these results are used for studying the spherical Fourier transform for the Gelfand pair \((G,K)\), \(G=SL_2({\mathbb C})\) and \(K=SU_2\). Here Gauss\((G)\) is the space generated by the \(K\)-biinvariant functions \[ \varphi _c(g)=e^{-{v^2\over 2c}}{v\over \sinh v}, \] where \(v\) is the geodesic distance from \(gK\) to the base point \(eK\) in the Riemannian symmetric space \(G/K\). By using the fact that \(q(x)=x\sinh x\) is admissible, the authors prove that Gauss\((G)\) is dense in \(L^1(K\backslash G/K)\), and as a result they obtain a new proof of the spherical Plancherel theorem for \(G=SL_2({\mathbb C})\).