半直线上的 de la Vallée Poussin 滤波多项式逼近
De la Vallée Poussin filtered polynomial approximation on the half line
AI总结：
本文在半直线上提出一种基于 de la Vallée Poussin 滤波器且易于计算的新型近最佳一致逼近多项式，证明了其 Lebesgue 常数一致有界且收敛速率近最佳，数值实验验证了其优于截断 Lagrange 插值的逼近性能及对 Gibbs 现象的有效抑制。
摘要：
在半直线上，我们引入了一组新的近最佳一致逼近多项式序列，该序列可通过被逼近函数在截断数量的 Laguerre 零点处的取值轻松计算得出。此类逼近多项式源自对经过 de la Vallée Poussin (VP) 滤波器滤波的 Fourier--Laguerre 部分和的离散化。它们的特点是依赖于两个参数：一个截断参数，决定了所考虑的 $n$ 个 Laguerre 零点中的数量；以及一个局部化参数，决定了我们将要应用的 VP 滤波器的作用范围。当 $n\to\infty$ 时，在这些参数及所涉及权重的 Laguerre 指数的简单假设下，我们证明，对于半轴上的任意局部连续函数，新的 VP 滤波逼近多项式具有一致有界的 Lebesgue 常数，并以近最佳的逼近速率一致收敛。\newline 理论结果已通过数值实验得到验证。具体而言，实验表明，在相同节点处，所提出的 VP 滤波逼近相较于截断 Lagrange 插值表现出更好的性能，特别是对于几乎处处非常平滑但具有孤立奇点的函数。在这些情况下，我们观察到了更具局部化的逼近，以及对 Gibbs 现象的良好抑制。
英文摘要：
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.
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