关于多元解析函数的切比雪夫多项式逼近的指数收敛性
On exponential convergence of Chebyshev polynomial approximation for multivariate analytic functions
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中文总结 AI 辅助
研究多元解析函数切比雪夫投影的指数收敛性，借助多势理论证明其在特定空间与最佳逼近有相同收敛速度，还扩展到相关主题并建立收敛速度，通过数值实验验证理论。
中文摘要 AI 辅助
本文利用多势理论对多元解析函数的切比雪夫投影进行了新的分析。证明了在任何向下封闭的凸多项式空间中，切比雪夫投影与最佳多项式逼近具有相同的指数收敛速度。该结果能够精确量化切比雪夫投影的指数收敛速度。分析扩展到几个相关主题，包括张量积切比雪夫插值、张量积高斯-勒让德求积、帕多瓦插值和求积以及切比雪夫-伽辽金方法，并在每种情况下建立了相应的指数收敛速度。提供了支持性数值实验以验证理论结果。
英文摘要
This paper presents a new analysis of the Chebyshev projection for multivariate analytic functions, drawing on pluripotential theory. It is proved that in any downward closed convex polynomial space, the Chebyshev projection achieves the same exponential convergence rate as the best polynomial approximation. This result enables a precise quantification of the exponential convergence rate of the Chebyshev projection. The analysis is then extended to several related topics, including tensorized Chebyshev interpolation, tensor product Gauss--Legendre quadrature, Padua interpolation and cubature, and Chebyshev-Galerkin method, with the corresponding exponential convergence rate established in each case. Supporting numerical experiments are provided to validate the theoretical results.
