发表机构
University College London（伦敦大学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文指出Wang关于解析函数Laguerre逼近根指数收敛证明的不完整性，提出用控制收敛定理及一个未证猜想补全证明，并基于数值实验认为猜想合理。
AI 中文摘要
在最近的一篇论文中（H. Wang, Math. Comp. 93, 2861-2884, 2024），Wang提出了若干关于基于广义Laguerre多项式的逼近收敛性的结果，声称提供了“解析函数Laguerre逼近根指数收敛的第一个严格证明”。在本注记中，我们认为Wang论文中主要结果的证明是不完整的，因为它们依赖于在积分号下取极限，而这并未得到恰当证明。我们解释了如何利用控制收敛定理来完成这些证明，前提是某些涉及广义Laguerre多项式及其加权Cauchy变换的非平凡不等式成立。我们提出了一个相关猜想，如果该猜想成立，将能完成Wang结果的证明。该猜想尚未被证明，但根据数值实验，它似乎是合理的。
英文摘要
In a recent paper (H. Wang, Math. Comp. 93, 2861-2884, 2024), Wang has presented a number of results concerning the convergence of approximations based on generalised Laguerre polynomials, claiming to have provided "the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions". In this note we argue that the proofs of the main results of Wang's paper are incomplete, because they rely on taking a limit under an integral sign, and this is not properly justified. We explain how the proofs could be completed using the dominated convergence theorem, provided that certain nontrivial inequalities involving generalised Laguerre polynomials and their weighted Cauchy transforms hold. We make a related conjecture, which, if true, would complete the proofs of Wang's results. This conjecture remains unproven, but appears plausible, based on numerical experiments.
