  The Fourier spectrum is a family of dimensions that interpolates between the
Fourier and Hausdorff dimensions and are defined in terms of certain energies
which capture Fourier decay. In this paper we obtain a convenient discrete
representation of those energies using the Fourier coefficients. As an example
application, we use this representation to establish sharp bounds for the
Fourier spectrum of a general measure with bounded support, improving previous
estimates of the second-named author
