  We consider the model $Z_i=X_i+\varepsilon_i$, for i.i.d. $X_i$'s and
$\varepsilon_i$'s and independent sequences $(X_i)_{i\in{\mathbb{N}}}$ and
$(\varepsilon_i)_{i\in{\mathbb{N}}}$. The density $f_{\varepsilon}$ of
$\varepsilon_1$ is assumed to be known, whereas the one of $X_1$, denoted by
$g$, is unknown. Our aim is to estimate linear functionals of $g$, $<\psi,g>$
for a known function $\psi$. We propose a general estimator of $<\psi,g>$ and
study the rate of convergence of its quadratic risk as a function of the
smoothness of $g$, $f_{\varepsilon}$ and $\psi$. Different contexts with
dependent data, such as stochastic volatility and AutoRegressive Conditionally
Heteroskedastic models, are also considered. An estimator which is adaptive to
the smoothness of unknown $g$ is then proposed, following a method studied by
Laurent et al. (Preprint (2006)) in the Gaussian white noise model. We give
upper bounds and asymptotic lower bounds of the quadratic risk of this
estimator. The results are applied to adaptive pointwise deconvolution, in
which context losses in the adaptive rates are shown to be optimal in the
minimax sense. They are also applied in the context of the stochastic
volatility model.
