Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight

Let there be given a partition of the closed interval $[-1,1]$ by arbitrary nodes $\{\eta_j\}_{j=0}^N$, where $\lambda_N=\max_{0\le j \le N-1} (\eta_{j+1}-\eta_{j})$. For a continuous function $f(t)$ given on an arbitrary grid $\Omega_N=\{t_j \mid \eta_{j} \le t_j \le \eta_{j+1}\}_{j=0}^{N-1}$, the approximation properties of the discrete Fourier sums $\Lambda^{\alpha,\beta}_{n,N}(f,t)$ in polynomials $\widehat P^{\alpha,\beta}_{n, N} (t)$ are investigated in the case of nonnegative integer parameters $\alpha$, $\beta$; these polynomials are orthogonal to $\Omega_N$ with Jacobi weight $\kappa^{\alpha,\beta}(t)=(1-t)^{\alpha}(1+t)^{\beta}$. Given the restriction $n=O(\lambda_N^{-1/3})$ on the order of the Fourier sums, a pointwise estimate of the Lebesgue function $L^{\alpha,\beta}_{n, N}(t)$ is obtained; it depends on $n$ and the position of the point $t \in [-1,1]$:
$$ L^{\alpha,\beta}_{n,N}(t)=O\bigl[\ln{(n+1)}+ |\widehat P^{\alpha,\beta}_{n,N}(t)|+ |\widehat P^{\alpha,\beta}_{n+1,N}(t)|\bigr]. $$