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

✍ Scribed by William O. Bray

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 734 KB
• Volume: 135
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0009

✦ Synopsis

On a symmetric space X=GÂK of noncompact type, we consider the formulas

where 8 * 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 P * on C c (X ) in the case G has real rank one. This result extends and generalizes a result due to Strichartz on odddimensional 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.