De la Vallée Poussin filtered polynomial approximation on the half–line

On the half line, we introduce a new sequence of near-best uniform approximation polynomials, easily computable by the values of the approximated function at a truncated number of Laguerre zeros. Such approximation polynomials come from a discretization of filtered Fourier–Laguerre partial sums, which are filtered using a de la Vallée Poussin (VP) filter. They have the peculiarity of depending on two parameters: a truncation parameter that determines how many of the n Laguerre zeros are considered, and a localization parameter, which determines the range of action of the VP filter we will apply. As $n\to \infty$, under simple assumptions on such parameters and the Laguerre exponents of the involved weights, we prove that the new VP filtered approximation polynomials have uniformly bounded Lebesgue constants and uniformly convergence at a near–best approximation rate, for any locally continuous function on the semiaxis.

The numerical experiments have validated the theoretical results. In particular, they show a better performance of the proposed VP filtered approximation versus the truncated Lagrange interpolation at the same nodes, especially for functions a.e. very smooth with isolated singularities. In such cases, we see a more localized approximation and a satisfactory reduction of the Gibbs phenomenon.