Uniform approximation on [−1, 1] via discrete de la Vallée Poussin means

Abstract

Starting from the function values on the roots of Jacobi polynomials, we construct a class of discrete de la Vallee Poussin means, by approximating the Fourier coefficients with a Gauss---Jacobi quadrature rule. Unlike the Lagrange interpolation polynomials, the resulting algebraic polynomials are uniformly convergent in suitable spaces of continuous functions, the order of convergence being comparable with the best polynomial approximation. Moreover, in the four Chebyshev cases the discrete de la Vallee Poussin means share the Lagrange interpolation property, which allows us to reduce the computational cost.