We introduce a new procedure to select the optimal cutoff parameter for Fourier density estimators that leads to adaptive rate optimal estimators, up to a logarithmic factor. This adaptive procedure applies for different inverse problems. We illustrate it on two classical examples: deconvolution and decompounding, i.e. non-parametric estimation of the jump density of a compound Poisson process from the observation of n increments of length ∆ > 0. For this latter example, we first build an estimator for which we provide an upper bound for its L 2-risk that is valid simultaneously for sampling rates ∆ that can vanish, ∆ := ∆ n → 0, can be fixed, ∆ n → ∆ 0 > 0 or can get large, ∆ n → ∞ slowly. This last result is new and presents interest on its own. Then, we show that the adaptive procedure we present leads to an adaptive and rate optimal (up to a logarithmic factor) estimator of the jump density.