Approximation properties of discrete Fourier sums in polynomials orthogonal on non-uniform grids

Given two positive integers α and β, for arbitrary continuous function f(x) on the segment [-1,1] we construct disrete Fourier sums Sα,βn,N(f,x) on system polynomials {p^α,βk,N(x)}N-1k=0 forming an orthonormals system on any finite non-uniform set ΩN={xj}N-1j=0 of N points from segment [-1,1] with Jacobi type weight. The approximation properties of the corresponding partial sums Sα,βn,N(f,x) of order n≤N-1 in the space of continuous functions C[-1,1] are investigated. Namely, for a Lebesgue function in Lα,βn,N(x), a two-sided pointwise estimate of discrete Fourier sums with n=O(δ-1(λ+3)N), λ=max{α,β}, δN=max0≤j≤N-1Δtj is obtained. The problem of convergence of Sα,βn,N(f,x) to f(x) is also investigated. In particular, an estimate is obtained of the deviation of the partial sum Sα,βn,N(f,x) from f(x) for n=O(δ-1(λ+3)N), depending on n and the position of a point x in [-1,1].