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 μ ̂ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0, iR) then μ ̂ 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.