On the Lévy-Raikov-Marcinkiewicz theorem

Let μ be a finite nonnegative Borel measure. The classical Lévy-Raikov-Marcinkiewicz theorem states that if its Fourier transform ̂m can be analytically continued to some complex half-neighborhood of the origin containing an interval (0, iR) then ̂m admits analytic continuation into the strip {t: 0 < st < 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. © 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All rights reserved.