  In this paper, we consider the distribution of the supremum of non-stationary
Gaussian processes, and present a new theoretical result on the asymptotic
behaviour of this distribution. Unlike previously known facts in this field,
our main theorem yields the asymptotic representation of the corresponding
distribution function with exponentially decaying remainder term. This result
can be efficiently used for studying the projection density estimates, based,
for instance, on Legendre polynomials. More precisely, we construct the
sequence of accompanying laws, which approximates the distribution of maximal
deviation of the considered estimates with polynomial rate. Moreover, we
construct the confidence bands for densities, which are honest at polynomial
rate to a broad class of densities.
