The aim of this paper is to estimate the density f of a random variable X when one has access to independent observations of the sum of K ≥ 2 independent copies of X. We provide a constructive estimator, based on a suitable definition of the logarithm of the Fourier transform of the observations. We propose a new strategy for the data driven choice of the cutoff parameter. It is proven optimal and a numerical study illustrates the performances of the method. Moreover, we discuss the fact that the definition of the estimator as well as the adaptive procedure apply in a wider context than the one considered here.