Fourier-Legendre approximation of a probability density from discrete data (Q1874198)

The author describes a positive approximation to a probability density \(u\) on \([0,1]\) if only a finite number of its values affected by noise is available. The procedure starts with estimating a number of Legendre-Fourier coefficients \(\lambda_k=\int_0^1 u(x)L_j(x)dx\). Then the estimate of \(u\) is found by the maximum entropy method that consists in minimising \(\int_0^1 u(x)\log u(x)dx\) over the family of probability densities with the given Legendre-Fourier coefficients. The author provides a theoretical result describing the quality of estimation and an example of application of this procedure in data-smoothing in the numerical solution of an identification problem for Fokker-Planck equation.