  This paper deals with the problem of density estimation. We aim at building
an estimate of an unknown density as a linear combination of functions of a
dictionary. Inspired by Cand\`es and Tao's approach, we propose an
$\ell_1$-minimization under an adaptive Dantzig constraint coming from sharp
concentration inequalities. This allows to consider a wide class of
dictionaries. Under local or global coherence assumptions, oracle inequalities
are derived. These theoretical results are also proved to be valid for the
natural Lasso estimate associated with our Dantzig procedure. Then, the issue
of calibrating these procedures is studied from both theoretical and practical
points of view. Finally, a numerical study shows the significant improvement
obtained by our procedures when compared with other classical procedures.
