Abstract: Let \(T=\{t_0, t_1, \ldots, t_N\}\) and \(T_N=\{x_0, x_1, \ldots, x_{N-1}\},\) where \(x_j=(t_j+t_{j+1})/2\), \(j=0, 1, \ldots, N-1,\) are any system of different points from \([-1, 1].\) For arbitrary continuous function \(f(x)\) on the segment \([-1, 1]\) we construct Valle-Poussin type averages \(V_{n,m,N}(f,x)\) for discrete Fourier sums \(S_{n,N}(f,x)\) on system of polynomials \(\{\hat{p}_{k,N}(x)\}_{k=0}^{N-1}\) forming an orthonormals system on any finite non-uniform grids \(T_N=\{x_j\}_{j=0}^{N-1}\) with weight \(\Delta{t_j}=t_{j+1}-t_j.\) Approximation properties of the constructed averages \(V_{n,m,N}(f,x)\) of order \(n+m\leq{N-1}\) in the space of continuous functions \(C[-1, 1]\) are investigated. Namely, it is proved that the Vallee-Poussin averages \(V_{m,n,N}(f,x)\) for \(\frac{n}m\asymp1, n\leq\lambda\delta_N^{-\frac14} (\lambda>0), \delta_N=\max_{0\leq{j}\leq{N-1}}\Delta{t_j},\) are uniformly bounded as a family of linear operators acting in the space \(C[-1, 1].\) In addition, as a consequence of the obtained result the order of approximation of the continuous function \(f(x)\) by the Vallee-Poussin \(V_{n,m,N}(f,x)\) averages in space \(C[-1, 1]\) is established.
For citation: Nurmagomedov, A. A. and Shikhshinatova, M. M. Approximation Properties of Valle-Poussin Averages for Discrete Fourier Sums Bypolynomials Orthogonal on Arbitrary Nets, Vladikavkaz Math. J., 2025, vol. 27, no. 2, pp. 93-111 (in Russian). DOI 10.46698/q4030-9541-4914-r
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