Construction of effective randomized projective estimates for solutions of integral equations based on Legendre polynomials

Numerical-statistical projective estimates for solutions of integral equations are constructed and optimized using Legendre polynomials as motivated by the computational complexity of orthogonal expansions with an adapted weight. By applying analytical and corresponding numerical computations, the mean-square error is minimized as a function of the length of the projection expansion segment, while the sample size for the expansion coefficients is fixed. The proposed technique is successfully verified in a test problem close to the Milne one and is found to be more effective than the regularized expansion in terms of Laguerre polynomials.