In this note we consider the estimation of an unknown function $f$ for weakly dependent data ($\alpha$-mixing) in a general setting. Our contribution is theoretical: we prove that a wavelet hard thresholding estimator attains a sharp rate of convergence under the mean integrated squared error (MISE) over Besov balls without imposing too restrictive assumptions on the model. Applications are given for two types of inverse problems: the deconvolution density estimation and the density estimation in a GARCH-type model, both improve existing results in this dependent context.