  In this paper, we consider projection estimates for L\'evy densities in
high-frequency setup. We give a unified treatment for different sets of basis
functions and focus on the asymptotic properties of the maximal deviation
distribution for these estimates. Our results are based on the idea to
reformulate the problems in terms of Gaussian processes of some special type
and to further analyze these Gaussian processes. In particular, we construct a
sequence of excursion sets, which guarantees the convergence of the deviation
distribution to the Gumbel distribution. We show that the rates of convergence
presented in previous articles on this topic are logarithmic and construct the
sequences of accompanying laws, which approximate the deviation distribution
with polynomial rate.
