  The theory of adaptive estimation and oracle inequalities for the case of
Gaussian-shift--finite-interval experiments has made significant progress in
recent years. In particular, sharp-minimax adaptive estimators and exact
exponential-type oracle inequalities have been suggested for a vast set of
functions including analytic and Sobolev with any positive index as well as for
Efromovich--Pinsker and Stein blockwise-shrinkage estimators. Is it possible to
obtain similar results for a more interesting applied problem of density
estimation and/or the dual problem of characteristic function estimation? The
answer is ``yes.'' In particular, the obtained results include exact
exponential-type oracle inequalities which allow to consider, for the first
time in the literature, a simultaneous sharp-minimax estimation of Sobolev
densities with any positive index (not necessarily larger than 1/2), infinitely
differentiable densities (including analytic, entire and stable), as well as of
not absolutely integrable characteristic functions. The same adaptive estimator
is also rate minimax over a familiar class of distributions with bounded
spectrum where the density and the characteristic function can be estimated
with the parametric rate.
