Approximation Properties of Valle- Poussin Averages for Discrete Fourier Sums Bypolynomials Orthogonal on Arbitrary Nets

Let T = {t0, t1, . . . , tN} and TN = {x0, x1, . . . , xN−1}, where xj = (tj + tj+1)/2, j = 0, 1, . . . ,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 Vn,m,N(f, x) for discrete Fourier sums Sn,N(f, x) on system of polynomials {ˆpk,N(x)}N−1 k=0 forming an orthonormals system on any finite non-uniform grids TN = {xj}N−1 j=0 with weight
tj = tj+1−tj . Approximation properties of the constructed averages Vn,m,N(f, x) of order n+m 6 N − 1 in the space of continuous functions C[−1, 1] are investigated. Namely, it is proved that the Vallee-Poussin averages Vm,n,N(f, x) for n m ≍ 1, n 6  − 1 4 N ( > 0), N = max06j6N−1
tj , 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 Vn,m,N(f, x) averages in space C[−1, 1] is established.