Moebius inversion of Fourier transforms

(*) F(t) fo oh(u) cos tu du is (t) -2 F(u) cos tu du. In this paper we present a method of inverting (*) which uses no integration whatsoever. The method consists of an application of the Moebius inversion formula combined with a variation of the classical Poisson formula from Fourier analysis. The main result is contained in Theorem 3. (Added in proof. It has been called to our attention that a similar result was announced by R. J. Duffiu in the Bulletin of the American Mathematical Society, vol. 47(1941), p. 383.) TEOREM 3. If 1. O(U) of bounded variation on (0 u R)for every R > O,