Generalized spectral projections on symmetric spaces of noncompact type: Paley-Wiener theorems
On a symmetric space X= G/K of noncompact type, we consider the formulas (equation presented) Pλf(x) = (f*Φλ)(x), where Φλ is the spherical function on X. Taken together they represent, the synthesis and decomposition formulas for appropriate functions f on X in terms of joint eigenfunctions of the invariant differential operators on X. The focus of this paper is the characterization of the range of Pλ on C∞c(X) in the case G has 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 on X due to Helgason. As an application we provide a fairly general support result for the spherical mean operator On X. © 1996 Academic Press, Inc.
Bray, William O. "Generalized spectral projections on symmetric spaces of noncompact type: Paley–Wiener theorems." journal of functional analysis 135, no. 1 (1996): 206-232.
Journal of Functional Analysis