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

Given two positive integers $\alpha$ and $\beta$, for arbitrary continuous function $f(x)$ on the segment $[-1, 1]$ we construct disrete Fourier sums $S_{n,N}^{\alpha,\beta}(f,x)$ on system polynomials $\big\{\hat{p}_{k,N}^{\alpha,\beta}(x)\big\}_{k=0}^{N-1}$ forming an orthonormals system on any finite non-uniform set $\Omega_N=\{x_j\}_{j=0}^{N-1}$ of $N$ points from segment $[-1, 1]$ with Jacobi type weight. The approximation properties of the corresponding partial sums $S_{n,N}^{\alpha,\beta}(f,x)$ of order $n\leq{N-1}$ in the space of continuous functions $C[-1, 1]$ are investigated. Namely, for a Lebesgue function in $L_{n,N}^{\alpha,\beta}(x)$, a two-sided pointwise estimate of discrete Fourier sums with $n=O\Big(\delta_N^{-\frac{1}{(\lambda+3)}}\Big)$, $\lambda=\max\{\alpha, \beta\}$, $\delta_N=\max_{0\leq{j}\leq{N-1}}\Delta{t_j}$ is obtained. The problem of convergence of $S_{n,N}^{\alpha,\beta}(f,x)$ to $f(x)$ is also investigated. In particular, an estimate is obtained of the deviation of the partial sum $S_{n,N}^{\alpha,\beta}(f,x)$ from $f(x)$ for $n=O\Big(\delta_N^{-\frac{1}{(\lambda+3)}}\Big)$, depending on $n$ and the position of a point $x$ in $[-1, 1].$