  The extremal Fourier intensities are studied for stationary
Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We
calculate the probability distribution of the maximal intensity and find that,
generically, it does not coincide with the distribution of the integrated power
spectrum (i.e. roughness of the surface), nor does it obey any of the known
extreme statistics limit distributions. The Fisher-Tippett-Gumbel limit
distribution is, however, recovered in three cases: (i) in the non-dispersive
(white noise) limit, (ii) for high dimensions, and (iii) when only
short-wavelength modes are kept. In the last two cases the limit distribution
emerges in novel scenarios.
