De la Vallée Poussin filtered polynomial approximation on the half line

TL;DR

This work addresses accurate uniform approximation on the half-line for locally continuous functions that may grow or have singularities by developing de la Vallée Poussin filtered polynomials based on Laguerre bases. It introduces a discrete VP variant $V_n^m(w_\alpha)$ obtained via Gauss-Laguerre discretization, proves uniform boundedness of the operator under $m\sim n$ and near-best convergence in the weighted norm, and demonstrates its effectiveness through numerical experiments that show reduced Gibbs phenomena and superior local accuracy compared with truncated Lagrange interpolation in many cases. The results provide a computationally efficient framework for half-line approximation with controlled Lebesgue constants and strong pointwise performance, supported by both theory and experiments. The approach is particularly advantageous for functions with smooth regions interrupted by isolated singularities, where VP filtering improves localization of the approximation and mitigates oscillations.

Abstract

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 by 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 that we are going to apply. As $n\to\infty$, under simple assumptions on such parameters and on 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. \newline The theoretical results have been validated by the numerical experiments. 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 as well as a good reduction of the Gibbs phenomenon.

Figures (11)

• Figure 1: Plots of the fundamental polynomials $\ell_{n,k}(x)$ and $\Phi_{n,k}^m(x)$ for $\alpha=-0.5$, $n=15$, $k=7$, and $m=\lfloor \theta n\rfloor$
• Figure 2: Example \ref{['es1']}: Plots of $\tilde{e}^{VP}_{n,m}(f_1,x)$ and $\tilde{e}^{Lag}_{n+1}(f_1,x)$ for $n=620$, $m=186$.
• Figure 3: Example \ref{['es2']}: Plots of $\tilde{e}^{VP}_{n,m}(f_2,x)$ and $\tilde{e}^{Lag}_{n+1}(f_2,x)$ for $n=420$, $m=126$.
• Figure 4: Example \ref{['es3']}: Plots of $\tilde{e}^{VP}_{n,m}(f_3,x)$ and $\tilde{e}^{Lag}_{n+1}(f_3,x)$ for $n=420$ and $m=378$
• Figure 5: Example \ref{['es4']}: Plots of $\tilde{e}^{VP}_{n,m}(f_3,x)$ and $\tilde{e}^{Lag}_{n+1}(f_3,x)$ for $n=500$ and $m=150$.