[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85795-en":3,"doc-seo-85795-105":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},85795,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Nonlinear Approximation with Adaptive Dictionaries","Nonlinear approximation with adaptive dictionaries studies how recovery errors for classes of functions defined by integral operators can be analyzed via sparse approximations of the kernel. Instead of classical Kolmogorov widths and classical bilinear dictionaries, the work focuses on an adaptive dictionary determined by the kernel itself. The approach extends asymptotic error analysis for sampling recovery from single smoothness classes to collections induced by operator-generated function kernels, building on prior results for widths and entropy numbers.","arXiv :2607 . 10052v1 [math .NA] 11 Jul 2026  \nNonlinear approximation with adaptive  \ndictionaries  \nV. Temlyakov  \nAbstract  \nIt is well known that the study of the Kolmogorov widths of a function class, which is the image of the unit ball of the Lq space of an integral operator JK with the kernel K, is closely connected with the study of sparse approximations of the kernel K with respect to the classical bilinear dictionary. Recently, it was discovered that if instead of the Kolmogorov widths we study the errors of optimal linear sampling recovery of the same classes, then we need to study sparse approximations of the kernel K with respect to an adaptive dictionary, which is determined by the kernel K. In this paper we study this important problem of nonlinear approximation with respect to an adaptive dictionary. Also, in this paper we continue to develop the following general approach, which is related to the above nonlinear approximation problem. We study asymptotic behavior of the errors of sampling recovery not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths and very recently for the entropy numbers.  \n1 Introduction  \nThis paper is a followup to the recent author’s paper [32] . In this paper we continue to study approximation of the multivariate functions K (x, y), x = (x1 ,..., xd ), y = (y1 ,..., yd ) by linear combinations of functions of the form u(x)v (y) . In the case, when we can choose arbitrary functions u and v, it is a classical problem of best bilinear approximation. In the paper [32] it was  \npointed out that the problem of optimal linear recovery on function classes defined by an integral operator with the kernel K (x, y) is closely related to the problem of approximation of K by linear combinations of functions of the form u(x)v (y) with functions v(y) defined by the kernel K(x, y), namely, v (y) = K(z, y) with some z. This means that we approximate K(x, y) with respect to a dictionary, which is determined by the function K (x, y) itself. We call such a process – approximation with adaptive dictionaries (see below for more details) . In the paper [32] mostly the case d = 1, i.e. the case of functions K (x, y) of two variables, was studied. In this paper we focus on the general case d ≥ 1.  \nWe now proceed to the detailed presentation. Let (Ω,µ) be a probability space. By the Lp , 1 ≤ p \u003C ∞ , norm we understand  \n∥f∥p := ∥f∥Lp (Ω,µ) := 􀀒ZΩ |f|p dµ􀀓 1/p .  \nBy the L∞-norm we understand the uniform norm of continuous functions  \n∥f∥∞ := sup|f(ω)|  \nω∈Ω  \nand with some abuse of notation we occasionally write L∞ (Ω) for the space C(Ω) of continuous functions on Ω . We define the vector Lp-norm, p =(p1 ,..., pv ), of functions of v variables x = (x1 ,..., xv ) as  \n∥f (x) ∥p := ∥f (x) ∥ (p1 , ...,pv) := ∥f (x) ∥p1 , ...,pv := ∥···∥f (·, x2 ,..., xv) ∥p1 ···∥pv .  \nWe now introduce some concepts from nonlinear sparse approximation.  \nThe first example of sparse approximation with respect to redundant dictionaries was considered by E. Schmidt in [14], who studied the approximation of functions f (x, y) of two variables by bilinear forms,  \nm  \nXui (x)vi (y) ,  \ni=1  \nin L2 ([0 , 1]2 ) . In this case we use the following dictionary (bilinear dictionary)  \nΠ := {u(x)v (y) : u, v ∈ L2 ([0 , 1])} , (1.1)  \nwhere the functions u and v are functions of a single variable. This problem is closely connected with properties of the integral operator  \n(Jfg)(x) := Z01 f (x, y)g (y)dy  \nwith the kernel f (x, y) . E. Schmidt ([14]) gave an expansion (known as the Schmidt expansion)  \n∞  \nf (x, y) =X sj (Jf)ϕj (x)ψj (y) , (1.2)  \nj=1  \nwhere {sj (Jf)} is a nonincreasing sequence of singular numbers of Jf , i.e. sj (Jf ) := λj (JJf)1/2, where {λj (A)} is a sequence of eigenvalues of an operator A, and J is the adjoint operator to","cbCaibbOyZWyNc5O","https://ap.wps.com/l/cbCaibbOyZWyNc5O","pdf",373446,1,30,"English","en",105,"# Introduction\n## Adaptive dictionaries and bilinear approximation\n## Norms and sparse approximation framework\n## Greedy dictionaries and best m-term errors","[{\"question\":\"What is the main problem studied in this paper?\",\"answer\":\"The paper studies nonlinear approximation errors using an adaptive dictionary determined by the kernel of an integral operator, focusing on how sampling recovery performance can be analyzed through sparse kernel approximation.\"},{\"question\":\"How do adaptive dictionaries differ from classical bilinear dictionaries?\",\"answer\":\"Classical bilinear dictionaries use fixed building blocks, while adaptive dictionaries are constructed from the kernel itself, so the dictionary depends on K(x,y).\"},{\"question\":\"What new perspective does the paper take compared with prior work on Kolmogorov widths?\",\"answer\":\"Rather than analyzing Kolmogorov widths of the function class image, it analyzes optimal linear sampling recovery errors for those classes, which leads to studying sparse approximations with respect to an adaptive dictionary.\"}]",1784206324,76,{"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},"nonlinear-approximation-with-adaptive-dictionaries","",{"@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/nonlinear-approximation-with-adaptive-dictionaries/85795/",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 the main problem studied in this paper?","Question",{"text":75,"@type":76},"The paper studies nonlinear approximation errors using an adaptive dictionary determined by the kernel of an integral operator, focusing on how sampling recovery performance can be analyzed through sparse kernel approximation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do adaptive dictionaries differ from classical bilinear dictionaries?",{"text":80,"@type":76},"Classical bilinear dictionaries use fixed building blocks, while adaptive dictionaries are constructed from the kernel itself, so the dictionary depends on K(x,y).",{"name":82,"@type":73,"acceptedAnswer":83},"What new perspective does the paper take compared with prior work on Kolmogorov widths?",{"text":84,"@type":76},"Rather than analyzing Kolmogorov widths of the function class image, it analyzes optimal linear sampling recovery errors for those classes, which leads to studying sparse approximations with respect to an adaptive 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