Adaptive estimation of linear functionals by model selection

We propose an estimation procedure for linear functionals based on Gaussian model selection techniques. We show that the procedure is adaptive, and we give a non asymptotic oracle inequality for the risk of the selected estimator with respect to the $\mathbb{L}_p$ loss. An application to the problem of estimating a signal or its $r^{th}$ derivative at a given point is developed and minimax rates are proved to hold uniformly over Besov balls. Simulations are included to illustrate the performances of the procedure for the estimation of a function on a whole interval. Our method provides a pointwise adaptive estimator.