Approximation properties of Valle-Poussin averages for discrete Fourier sums bypolynomials orthogonal on arbitrary nets

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.