  This paper aims to build an estimate of an unknown density of the data with
measurement error as a linear combination of functions from a dictionary.
Inspired by the penalization approach, we propose the weighted Elastic-net
penalized minimal $\ell_2$-distance method for sparse coefficients estimation,
where the adaptive weights come from sharp concentration inequalities. The
optimal weighted tuning parameters are obtained by the first-order conditions
holding with a high probability. Under local coherence or minimal eigenvalue
assumptions, non-asymptotical oracle inequalities are derived. These
theoretical results are transposed to obtain the support recovery with a high
probability. Then, some numerical experiments for discrete and continuous
distributions confirm the significant improvement obtained by our procedure
when compared with other conventional approaches. Finally, the application is
performed in a meteorology data set. It shows that our method has potency and
superiority of detecting the shape of multi-mode density compared with other
conventional approaches.
