We describe a method for distribution function and density estimation with Bern- stein polynomials. We take advantage of results about the eigenstructure of the Bernstein operator to re ne the Sevy's convergence acceleration method, based on iterates of this opera- tor; the original Sevy's algorithm is improved by introducing fractional operators. The proposed algorithm has better convergence properties than the classical one; the price to pay is a control- lable loss of the shape-preserving properties of the Bernstein approximation (monotonicity and positivity in the Density Estimation setting). The method is tested on simulated data.