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 Vall\'ee 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.

翻译：在半直线上，我们引入了一种新的近最优一致逼近多项式序列，该序列可通过截断的拉盖尔零点处被逼近函数的值轻松计算。这类逼近多项式源于离散化的滤波傅里叶-拉盖尔部分和，这些部分和通过德·拉·瓦莱·普桑滤波器进行滤波。其独特之处在于依赖两个参数：一个截断参数，决定所考虑的 n 个拉盖尔零点中的多少个被采用；一个局部化参数，决定我们应用的 VP 滤波器的作用范围。当 n→∞ 时，在关于这些参数及所涉权函数的拉盖尔指数的简单假设下，我们证明新的 VP 滤波逼近多项式具有一致有界的勒贝格常数，并对半轴上的任意局部连续函数以近最优逼近率一致收敛。理论结果已通过数值实验验证。特别地，数值实验表明，与相同节点处的截断拉格朗日插值相比，所提出的 VP 滤波逼近性能更优，尤其对于具有孤立奇点的几乎处处非常光滑的函数。在此类情形中，我们观察到更局部化的逼近以及吉布斯现象的有效抑制。