ON THE PALEY-WIENER THEOREM

Abstract. The Fourier transforms of functions with compact and convex supports as well as with non-convex and unbounded supports are considered. Key words. Paley-Wiener Theorem, Fourier Transform Mathematics Subject Classifications. Primary 42B10 In this paper we describe real-variable characteristics of Fourier transforms of functions with compact supports in Rn. Real-variable descriptions of Fourier transforms of functions with nonconvex or unbounded supports are also considered. Theorem 1. Let f ∈ C∞(Rn) such that all Laplacians ∆kf, k = 0, 1,..., belong to L2(R n). Then there always exists the limit σf = lim k→∞ ‖∆kf‖1/(2k)2, (1) and moreover, σf = sup{|ξ | : ξ ∈ suppf̂(ξ)}, (2) where f̂(ξ) is the Fourier transform of the function f(x) (see [2]): f̂(ξ) = (2pi)−n/2 Rn f(x)eix.ξdx, (3) and the support of a function f is the smallest closed set such that f = 0 almost everywhere in its complement. The proof is based on the Plancherel theorem for the Fourier transform and the inequalities (σf − )4