On the Lévy-Raikov-Marcinkiewicz theorem

Abstract

Let μ be a finite non-negative Borel measure. The classical Lévy-Raikov-Marcinkiewicz theorem states that if its Fourier transform μ̂ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0,iR) then μ̂ admits analytic continuation into the strip {t: 0<It<R}. We extend this result to general classes of measures and distributions, assuming non-negativity only on some ray and allowing temperate growth on the whole line. © 2004 Elsevier Inc. All rights reserved.