1. One of the best known theorems on the finite Fourier transform is:—The integral function F(z) is of the exponential type C and belongs to L2 on the real axis, if and only if, there exists an f(x) belonging to L2 (—C, C) such that ( Additionally, if f(x) vanishes almost everywhere in a neighbourhood of C and also in a neighbourhood of —C, then F(z) is of an exponential type lower than C.
A note on the finite Fourier transform