The approximation of functions by partial sums of the Fourier series in polynomials orthogonal on arbitrary grids

For arbitrary continuous function $f(t)$ on the segment $[-1, 1]$ we construct discrete sums by Fourier $S_{n,N}(f,t)$ on system polynomials forming an orthonormals system on any finite non-uniform set $T_N = \{t_j\}_{j=0}^{N-1}$ of $N$ points from segment $[-1, 1]$ with weight $\Delta{t_j} = t_{j+1} - t_j.$ Approximation properties of the constructing partial sums $S_{n,N}(f,t)$ order $n\leq{N-1}$ are investiga-ted. Namely a two-sided pointwise estimate is obtained for the Lebesgue function $L_{n,N}(t)$ discrete Fourier sums for $n=O(\delta_N^{-1/5}), \delta_N=\max_{0\leq{j}\leq{N-1}}\Delta{t_j}$. Coherently also is investigated the question of the convergence of $S_{n,N}(f,t)$ to $f(t).$ In particular, we obtaine the estimation deflection partial sums $S_{n,N}(f,t)$ from $f(t)$ for $n=O(\delta_N^{-1/5})$ which is depended on $n$ and position of a point $t$ on the $[-1, 1].$