Local Paley-Wiener theorems for functions analytic on unit spheres.

The purpose of this paper is to provide new and simplified statements of local Paley-Wiener theorems on the (n? 1)-dimensional unit sphere realized as a subset of n= 2, 3 Euclidean space. More precisely, given a function , whose restriction to an n? 1 sphere is analytic, we establish necessary and sufficient conditions determining whether fis the Fourier transform of a compactly supported, bounded function . The essence of this investigation is that, because of the local nature of the problem, the mapping f? Fis not in general invertible and so the problem cannot be studied via a Fourier integral. Our proofs are new. [ABSTRACT FROM AUTHOR]