[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85849-en":3,"doc-seo-85849-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85849,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","On Exponential Convergence of Chebyshev Polynomial Approximation for Multivariate Analytic Functions","The paper analyzes Chebyshev projection for multivariate analytic functions using pluripotential theory. It proves that in any downward closed convex polynomial space, Chebyshev projection attains the same exponential convergence rate as the best polynomial approximation, enabling an exact quantification of this rate. The framework is extended to tensorized Chebyshev interpolation, tensor product Gauss–Legendre quadrature, Padua interpolation and cubature, and the Chebyshev–Galerkin method, with exponential convergence rates established in each setting, supported by numerical experiments.","arXiv :2607 . 10209v1 [math .NA] 11 Jul 2026  \nOn exponential convergence of Chebyshev polynomial approximation for multivariate analytic functions  \nXinyu Wang∗ and Haiyong Wang∗†  \nJuly 14, 2026  \nAbstract  \nThis 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.  \nKeywords: Chebyshev polynomial approximation, multivariate analytic functions, exponential convergence, pluripotential theory, Bernstein–Walsh theorem  \nAMS classifications: 41A10, 41A63, 41A25  \n1 Introduction  \nChebyshev polynomial approximations play a crucial role in many areas of scientific computing, including spectral methods for solving PDEs [6, 9, 23], the Chebfun software for numerical computing [10], option pricing and finance modeling [11, 12], deep neural networks [26], etc. Consider a function f(x) defined on the d-dimensional hypercube [−1, 1]d for some d ∈ N, and let Tk(x) = cos(k arccos(x)) denote the Chebyshev polynomial of the first kind of degree k. If f satisfies the Dini–Lipschitz condition [18, Theorem 4.1], then it has a uniformly and absolutely Chebyshev expansion of the form  \n∞ ∞  \nf (x) = X akTk(x) = X ···X ak1 , ...,kdTk1 (x1 )···Tk d (xd), (1.1)  \nk∈Nd0 k1=0 k d=0  \n∗ School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan  \n430074, [P. R. China.](P. R. China. Email:haiyongwang@hust.edu.cn)[ Email:haiyongwang@hust.edu.cn](P. R. China. Email:haiyongwang@hust.edu.cn)  \n†Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, P. R. China.  \nwhere Nd0 denotes the set of all d-tuples of nonnegative integers. Truncating this expansion yields the Chebyshev projection of f:  \nSΛ(f)(x) =X akTk(x),  \nk∈Λ  \nwhere Λ ⊂ Nd0 is a finite multi-index set chosen suitably. In the univariate case with Λ = {0,..., n}, the exponential convergence of the Chebyshev projection for analytic functions was first established by Bernstein in 1912: if f is analytic in a neighbourhood of E (ρ), where E (ρ) denotes the open region bounded by the Bernstein ellipse  \n∂E (ρ) := 􀀚 z ∈ C : z = u~~ ~~+2u−1 , |u| = ρ 􀀛 , ρ > 1 ,  \nthen max x∈[−1 , 1] |f(x) − SΛ(f)(x)| = O (ρ−n ) [2, pp. 94-95] . The parameter ρ can be explicitly determined from the locations of the singularities of f. In the multivariate case, however, only a few discussions have been devoted to analyzing the exponential convergence of Chebyshev projection. Key contributions can be summarized as follows:  \n• Bochner and Martin in [3, Chapter V] established a multivariate analogue of Bernstein’s result, showing that if f is analytic in a neighbourhood of E (ρ1 ) ×···×E(ρd) , then its Chebyshev coefficients satisfy ak = O (ρ−1k1 ··· ρ−dk d) for k = (k1 , ... , kd) ∈ Nd0 . Although the exponential convergence behavior of Chebyshev projection can be easily seen, the quantification of the exponential convergence rate remains open.  \n• More recently, based on the observation that multivariate polynomials of fixed degree have anisotropic resolution power in the hypercube, Trefethen in [28,29] introduced the Euclidean degree for multivariate polynomials, and quantified explicitly the exponential convergence rates of Chebyshev projections with total, Euclidean and maximal degrees. Altho","cbCaimOmayqJP6to","https://ap.wps.com/l/cbCaimOmayqJP6to","pdf",927609,3,1,26,"English","en",105,"# Introduction\n# Main Results and Pluripotential-Theoretic Analysis\n## Chebyshev Projection in Downward Closed Convex Polynomial Spaces\n## Quantifying Exponential Convergence Rate\n# Extensions to Related Approximation Schemes\n## Tensorized Chebyshev Interpolation\n## Gauss–Legendre Quadrature\n## Padua Interpolation and Cubature\n## Chebyshev–Galerkin Method\n# Numerical Experiments","[{\"question\":\"What is the main theoretical contribution regarding Chebyshev projection?\",\"answer\":\"The paper proves that, within any downward closed convex polynomial space, the Chebyshev projection achieves the same exponential convergence rate as the best polynomial approximation.\"},{\"question\":\"How is the exponential convergence rate for multivariate analytic functions quantified?\",\"answer\":\"The analysis uses pluripotential theory to derive upper bounds for Chebyshev coefficients based on analyticity assumptions tied to the hypercube’s P-extremal function, yielding a precise maximal convergence characterization.\"},{\"question\":\"Which related numerical schemes are covered beyond basic Chebyshev projection?\",\"answer\":\"The results extend to tensorized Chebyshev interpolation, tensor product Gauss–Legendre quadrature, Padua interpolation and cubature, and the Chebyshev–Galerkin method, each with an established exponential convergence rate.\"}]",1784206686,66,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"on-exponential-convergence-of-chebyshev-polynomial-approximation-for-multivariate-analytic-functions","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/on-exponential-convergence-of-chebyshev-polynomial-approximation-for-multivariate-analytic-functions/85849/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main theoretical contribution regarding Chebyshev projection?","Question",{"text":75,"@type":76},"The paper proves that, within any downward closed convex polynomial space, the Chebyshev projection achieves the same exponential convergence rate as the best polynomial approximation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the exponential convergence rate for multivariate analytic functions quantified?",{"text":80,"@type":76},"The analysis uses pluripotential theory to derive upper bounds for Chebyshev coefficients based on analyticity assumptions tied to the hypercube’s P-extremal function, yielding a precise maximal convergence characterization.",{"name":82,"@type":73,"acceptedAnswer":83},"Which related numerical schemes are covered beyond basic Chebyshev projection?",{"text":84,"@type":76},"The results extend to tensorized Chebyshev interpolation, tensor product Gauss–Legendre quadrature, Padua interpolation and cubature, and the Chebyshev–Galerkin method, each with an established exponential convergence rate.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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