There is a relation between the Paley-Wiener theorem and an earlier representation due to E. Borel.
(b) The connection with Theorem 3.3 is as follows. For these functions f (for which in addition f and are of moderate decrease on the real axis), one can assert that the g above is holomorphic in the larger region, which consists of the slit plane C − [−M,M]. Moreover, the relation between g and the Fourier transform is
where h is continuous and supported in [−M,M].
(a) Prove that the function g is holomorphic in C − [−M,M], and vanishes at infinity, that is, lim|z|→∞ |g(z)| = 0. Moreover, the “jump” of g across [−M,M] is h, that is,
[Hint: If G is another function satisfying these conditions, g − G is entire.]