A Discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmetric spaces

Let $f$ be a Paley-Wiener function in the space $L_2(X)$, where $X$ is a symmetric space of noncompact type. It is shown that by using the values of $f$ on a sufficiently dense and separated set of points of $X$ one can give an exact formula for the Helgason-Fourier transform of $f$. In order to find a discrete approximation to the Helgason-Fourier transform of a function from a Besov space on $X$ we develop an approximation theory by Paley-Wiener functions in $L_2(X)$.