Paley-Wiener theorem for the Joint-Eigenspace Fourier transform on noncompact symmetric spaces

Abstract

This paper conducts a geometric analysis of the Joint-Eigenspace Fourier transform of the symmetric space of the non-compact type. Our study shows how the Poisson transform builds up the well-known Helgason Fourier transform for an analysis of the complete duality of the underlying symetric space. Among other results, we establish an inversion formula, a Plancherel formula and (as our main result) a Paley-Wiener theorem for the Joint-Eigenspace Fourier transform on any noncompact symmetric space.

This paper considers another aspect of the duality of the symmetric spaces of the noncompact type in which the Poisson transform builds up the Helgason Fourier transform into the Joint-Eigenspace Fourier transform, thus leading to a complete duality of these spaces. Comments are appreciated