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

TL;DR

This paper develops a Paley-Wiener theorem for the Joint-Eigenspace Fourier transform on noncompact symmetric spaces, providing foundational analysis tools for understanding harmonic analysis in these geometric settings.

Contribution

It introduces a Paley-Wiener theorem for the Joint-Eigenspace Fourier transform on noncompact symmetric spaces, extending classical Fourier analysis results.

Findings

Established an inversion formula for the transform

Derived a Plancherel formula for the transform

Proved a Paley-Wiener theorem for the joint-eigenspace Fourier transform

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.