Abstract
Let Ω = {t0, t1, …, tN} and ΩN = {x0, x1, …, xN–1}, where xj = (tj + tj + 1)/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums Sn, N( f, x) with respect to the system of polynomials {p̂k,N(x)} N–1 k=0 , forming an orthonormal system on nonuniform point systems ΩN consisting of finite number N of points from the segment [–1, 1] with weight Δtj = tj + 1–tj. We find the growth order for the Lebesgue function Ln,N (x) of the considered partial discrete Fourier sums Sn,N ( f, x) as n = O(δ −2/7 N ), δN = max0≤ j≤N−1 Δtj More exactly, we have a two-sided pointwise estimate for the Lebesgue function Ln, N(x), depending on n and the position of the point x from [–1, 1].
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Original Russian Text © A.A. Nurmagomedov, N.K. Rasulov, 2018, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2018, Vol. 63, No. 3, pp. 417–430.
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Nurmagomedov, A.A., Rasulov, N.K. Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids. Vestnik St.Petersb. Univ.Math. 51, 249–259 (2018). https://doi.org/10.3103/S1063454118030068
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DOI: https://doi.org/10.3103/S1063454118030068