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 f : C-n -> C, n = 2, 3, whose restriction to an n - 1 sphere is analytic, we establish necessary and sufficient conditions determining whether f is the Fourier transform of a compactly supported, bounded function F : R-n -> C. The essence of this investigation is that, because of the local nature of the problem, the mapping f -> F is not in general invertible and so the problem cannot be studied via a Fourier integral. Our proofs are new.