Approximate properties of the de la Vallée Poussin means for the discrete Fourier–Jacobi sums

We consider the system of the classical Jacobi polynomials of degree at most $N$ which generate an orthogonal system on the discrete set of the zeros of the Jacobi polynomial of degree $N$. Given an arbitrary continuous function on the interval $[-1,1]$, we construct the de la Vallée Poussin-type means for discrete Fourier–Jacobi sums over the orthonormal system introduced above. We prove that, under certain relations between $N$ and the parameters in the definition of de la Vallée Poussin means, the latter approximate a continuous function with the best approximation rate in the space $C[-1,1]$ of continuous functions.