[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84739-en":3,"doc-seo-84739-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84739,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Smoothing by, and eccentric smoothing of, compactly supported RBFs","We consider compactly supported radial basis functions (RBFs) whose Fourier transforms decay algebraically. The focus is on generalized Wendland RBFs and a modification called eccentric smoothing, which increases smoothness away from the origin. The work analyzes mapping properties of the associated integral (covariance) operators for these kernels. It further shows that eccentric smoothing enables wavelet-inspired compression of the kernel matrix, making sparse representations practical.","arXiv :2607 .04512v1 [math .NA] 5 Jul 2026  \nSmoothing by, and eccentric smoothing of, compactly supported RBFs  \nT. Hangelbroek∗, C. Rieger†  \nJuly 7, 2026  \nAbstract  \nWe consider compactly supported RBFs having algebraically decaying Fourier transforms. Here we focus especially on generalized Wendland RBFs and their modification by making them smoother away from zero, a process we call eccentric smoothing. Specifically, we consider mapping properties of the integral operators (sometimes known as covariance operator) for these novel type RBFs. Moreover, we show that eccentric smoothing makes wavelet-inspired compression technique for the kernel matrix feasible.  \n1 Introduction  \nA compactly supported RBF is radially symmetric function ϕ : Rd → R, with suppϕ ⊂ B(0, 1) which is positive definite: meaning that for any finite set X = {x1 ,... xn }, the collocation matrix ΦX =􀀀ϕ(xj − xk)􀀁 j,k≤n is strictly positive definite.  \nIn this article, we consider compactly supported RBFs which have have finite global smoothness ϕ ∈ C λ1 (Rd ) . Our main object of study is to consider the (higher) smoothness away from the origin: ϕ ∈ C λ2 􀀀Rd \\{0} 􀀁 for some λ2 > λ 1 , possibly infinity. When λ2 is finite, we are especially interested in the case that it is sharp (so that ϕ  C λ2 +ϵ 􀀀Rd \\ {0} 􀀁 for ϵ > 0) . This functions as a useful secondary parameter for the RBF, which we call the eccentric smoothing parameter.  \nThe advantage of an increased eccentric smoothing parameter was noted in [10], where it was used to obtaining high order convergence rates in high order Sobolev norms for approximation and interpolation with compactly supported RBFs. There are other advantages.  \nCompression of the collocation matrix Recently, the compressibility of the matrix ΦX , which is full (despite the compact support of ϕ) and becomes ill-conditioned for large or poorly arranged point sets X , has been a favorable outcome of the study of samplets.  \nFor certain RBFs, after a change of basis (represented by the matrix T), the collocation matrix ΦΣX = T ΦXT−1 can be rendered sparse with little cost. The ability to compress the matrix relies on the smoothness of the kernel away from the origin.  \nSpecifically, it relies on the asymptotic smoothness of the kernel. This is a concept which has migrated from the theory of compression of H matrices [9, Section 4.3.4] . The original condition is very strong condition and compactly supported functions cannot satisfy it. However, a weaker version, namely asymptotic smoothness of finite order, has been considered for positive definite kernels, where it is key to producing wavelet-like bases (i.e., samplets) for kernel spaces in [12, 13] . Indeed, for globally supported RBFs, this type of compression has been used in [2] to produce an effective multiscale algorithm for RBF interpolation with computational cost O (N log N), where N = Ξ , and Ξ is assumed to be quasi-uniform. This notion of asymptotic smoothness provides sufficient conditions for compressibility of the transformed collocation matrix – we expect that the smoothness away from the origin provided by Proposition 3.1 will allow compactly supported kernels to be used effectively in this setting.  \n∗ Department of Mathematics,University of Hawai‘i – M¯anoa  \n†Department of Mathematics and Computer Science, Philipps-Universität Marburg  \n2020 Mathematics Subject Classification. 65D12, 65F55, 47G30 .  \nKey words and phrases. Compactly supported RBFs , Samplet compression, Pseudodifferential operators, Smoothing integral operator .  \nRegularity of the integral operator The integral operator of ϕ plays an important role in kernel approximation, machine learning and statistics [7, 15, 4], the square root of the operator acting on L 2 (σ) is the kernel’s “native space”, the eigenvectors of the operator are fundamental to Mercer and KarhunenLoève expansions (see [7, Ch. 3] [4, Ch. 2.3] for background) . The range of the operator on L 2 (σ) is the doubling class co","cbCaifNdW8PZ9F7B","https://ap.wps.com/l/cbCaifNdW8PZ9F7B","pdf",476648,1,9,"English","en",105,"# Introduction\n# Compression of the collocation matrix\n# Regularity of the integral operator\n# Outline","[{\"question\":\"What is eccentric smoothing in the context of compactly supported RBFs?\",\"answer\":\"Eccentric smoothing modifies generalized Wendland RBFs to become smoother away from the origin while remaining compactly supported.\"},{\"question\":\"Why is smoothness away from the origin important for matrix compression?\",\"answer\":\"Increased smoothness away from zero supports asymptotic smoothness of the kernel, which underpins compressibility after a change of basis (samplet/wavelet-inspired constructions).\"},{\"question\":\"What challenges arise for the integral operator when the eccentric smoothing regularity parameter is finite?\",\"answer\":\"When the away-from-origin smoothness parameter is finite, the Fourier symbol does not fall into Hörmander pseudodifferential symbol classes, making mapping properties on Sobolev and related spaces more difficult.\"}]",1784197971,23,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"smoothing-by-and-eccentric-smoothing-of-compactly-supported-rbfs","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/smoothing-by-and-eccentric-smoothing-of-compactly-supported-rbfs/84739/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 eccentric smoothing in the context of compactly supported RBFs?","Question",{"text":75,"@type":76},"Eccentric smoothing modifies generalized Wendland RBFs to become smoother away from the origin while remaining compactly supported.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is smoothness away from the origin important for matrix compression?",{"text":80,"@type":76},"Increased smoothness away from zero supports asymptotic smoothness of the kernel, which underpins compressibility after a change of basis (samplet/wavelet-inspired constructions).",{"name":82,"@type":73,"acceptedAnswer":83},"What challenges arise for the integral operator when the eccentric smoothing regularity parameter is finite?",{"text":84,"@type":76},"When the away-from-origin smoothness parameter is finite, the Fourier symbol does not fall into Hörmander pseudodifferential symbol classes, making mapping properties on Sobolev and related spaces more 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