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 Ω N = = {t j } N −1 j=0 , where nodes t j satisfy the only condition η j t j η j+1 , 0 j N − 1, and nodes η j are such that −1 = η 0 < η 1 < η 2 < < • • • < η N −1 < η N = 1. We investigate approximative properties of the finite Fourier series for f (t) by algebraic polynomialsP n, N (t), that are orthogonal on Ω N = {t j } N −1 j=0. Lebesgue-type inequalities for the partial Fourier sums byP n, N (t) are obtained.