ON THE CONVERGENCE OF THE LEAST SQUARE METHOD IN CASE OF NON-UNIFORM GRIDS

Abstract

Let f(t) be a continuous on [−1, 1] function, which values are given at the points of arbitrary non-uniform grid ΩN = = {tj} N−1 j=0 , where nodes tj satisfy the only condition ηj 6tj 6ηj+1, 0 6 j 6 N − 1, and nodes ηj are such that −1 = η0 < η1 < η2 < < · · · < ηN−1 < ηN = 1. We investigate approximative properties of the finite Fourier series for f(t) by algebraic polynomials Pˆ n, N (t), that are orthogonal on ΩN = {tj} N−1 j=0 . Lebesgue-type inequalities for the partial Fourier sums by Pˆ n, N (t) are obtained.

Keywords

• random net
• non-uniform grid
• orthogonal polynomials
• legendre polynomials
• least square method
• fourier series
• function approximation