Near best discrete polynomial approximation via de la Vallee Poussin means

One of the most popular discrete approximating polynomials is the Lagrange interpolation polynomial and the Jacobi zeros provide a particularly convenient choice of the interpolation knots on [?1, 1]. However, it is well known that there is no point system such that the associate sequence of Lagrange polynomials, interpolating an arbitrary function f, would converge to f w.r.t. any weighted uniform or L1 norm. To overcome this problem, some discrete approximating polynomials have been originated from certain delayed arithmetic means of the Fourier-Jacobi partial sums (de la Vallee Poussin means) by approximating the Fourier coefficients with a Gaussian quadrature rule. The uniform convergence of these polynomials in suitable spaces of continuous functions has been recently proved. In this talk we complete the study by analyzing the approximation error w.r.t. the weighted L1 norm. In the main estimate we state, we use Ditzian-Totik moduli of smoothness.