Double-sided to Lebesgue function of Fourier sums by polynomials orthogonal on non-uniform grids

Abstract

Let Ω = {t0, t1, . . . , tN} and ΩN = {x0, x1, . . . , xN−1}, where xj = (tj + tj+1)/2, j = 0, 1, . . . , N − 1 are any systems of different points from [−1, 1]. For arbitrary continuous function f(x) on the segment [−1, 1] discrete Fourier sums is constructed by a system of polynomials {ˆpk,N(x)}N−1 k=0 forming an orthonormal system on non-uniform system of points ΩN consisting of a finite number of N points from segment [−1, 1] with weight Δtj = tj+1 − tj . We have found an order for Lebesgue function Ln,N(x) of the growing discrete Fourier sums Sn,N(f, x) for n = O(δ −2/7 N ), δN = max0 ≤j≤N−1 Δtj. Namely, a double-sided pointwise estimate for Lebesgue function Ln,N(x) with is depended from n and position of a point x on the [−1, 1] is obtained.