Generalized Spectral Projections on Symmetric Spaces of Noncompact Type: Paley–Wiener Theorems

On a symmetric spaceX=G/Kof noncompact type, we consider the formulas[formula][formula]whereΠλis the spherical function onX. Taken together they represent, the synthesis and decomposition formulas for appropriate functionsfonXin terms of joint eigenfunctions of the invariant differential operators onX. The focus of this paper is the characterization of the range ofPλonCc∞(X) in the caseGhas real rank one. This result extends and generalizes a result due to Strichartz on odd-dimensional real hyperbolic space and provides a “spectral” reformulation of the Paley–Wiener theorem for Fourier transform onXdue to Helgason. As an application we provide a fairly general support result for the spherical mean operator onX.