Fourier–Legendre approximation of a probability density from discrete data

We produce a positive approximation of a probability density in [0,1] when only a finite number of values (possibly affected by noise) is available. This approximation is obtained by computing a number of Legendre–Fourier coefficients and applying the Maximum Entropy method. An example of application of this procedure is data-smoothing in the numerical solution of an identification problem for Fokker–Planck equation.