Past exam of the real and harmonic analysis course of the University of Oxford / 2008 / 3 / f / Solution

Convolution preserves probability measures: its defining pushforward is nonnegative and its total mass is the product of the two unit masses. For every fixed integer $n$ the previous coefficient identity givesThe probability version of the Fourier coefficient criterion therefore provesThe fixed compact underlying group and the uniform probability masses supply precisely the bounds needed in the dense-test argument.