  The main result of this paper is, that if we suppose that a function is
absolutely continuous and uniformly H\"older continuous and that it's finite
difference function does not oscillate infinitely often on a bounded interval,
then the decay rate of its Fourier coefficients can be estimated exactly. This
rate of decay predicts the same uniform H\"older continuity but the two other
conditions are not necessary. Several examples from literature and by the
author show that none of the assumptions can be relaxed without weakening the
decay for some functions. The uniform H\"older continuity of chirps and the
decay of their Fourier coefficients are studied. The main result is then
applied in the estimation of the error of numerical Weyl fractional derivatives
calculated using the discrete Fourier transform. The main result is also
extended to Fourier transforms.
