[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83626-en":3,"doc-seo-83626-105":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},83626,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Weighted Derivative Histopolation on Arbitrary Grids: Admissibility and Exact Factorizations","A weighted derivative histopolation framework reconstructs polynomials from weighted integral moments of the derivative over prescribed subinterval families. The degrees of freedom combine one scalar normalization with weighted moment data. Unisolvence on ΠN is proved under conditions that the interval family separates polynomials up to degree N−1 and the normalization is nonzero on constants. The paper provides a sharp well-posedness criterion and describes admissible normalizations. For admissible interval families derived from a fixed grid, admissibility reduces to nonsingularity of an associated interval matrix, with special block/diagonal structures under Jacobi and Chebyshev settings, plus exact sine-transform factorizations in selected configurations, supported by numerical experiments.","arXiv :2607 .02105v1 [math .NA] 2 Jul 2026  \nNoname manuscript No.  \n(will be inserted by the editor)  \nWeighted Derivative Histopolation on Arbitrary Grids: Admissibility and Exact Factorizations  \nAllal Guessab · Federico Nudo  \nReceived: date / Accepted: date  \nAbstract In this paper, we introduce a weighted derivative histopolation framework on families of intervals. The degrees of freedom consist of one scalar normalization and weighted integral moments of the derivative over a prescribed family of subintervals. We prove that the resulting scheme is unisolvent on ΠN when the interval family separates polynomials of degree at most N−1 through weighted moments and the normalization is nonzero on constants. Thus, the derivative moments determine the polynomial up to an additive constant, and the scalar normalization fixes this remaining degree of freedom. This gives a sharp criterion for the well-posedness of the interpolation problem and a complete characterization of the admissible scalar normalizations. We then show how admissible families of intervals can be constructed from a fixed grid. When the endpoints of the intervals belong to the grid, admissibility is reduced to the nonsingularity of an interval matrix associated with the family, which depends only on the representation of the intervals in terms of consecutive cells. For Jacobi weights, the associated data matrices have a natural block structure in Jacobi polynomial bases, and the reduced derivative matrix can be expressed in terms of shifted Jacobi moment matrices. We next study Chebyshev configurationsin which this structure becomes explicit. For the four classical Chebyshev families, suitable polynomial bases lead to diagonal Gram matrices for the reduced derivative matrices. We show that this diagonal structure depends on the simultaneous choice of the weight, the basis, and the grid. In particular, for general Jacobi weights on a fixed Chebyshev–Lobatto grid, the corresponding discrete orthogonality is not available in general, although an identity Gram matrix can always be obtained by a change of basis. We also identify configurations in which the reduced derivative matrices admit exact sine transform factorizations, yielding explicit singular values and spectral condition numbers. Numerical experiments on equispaced and Chebyshev–Lobatto nodes show the behaviour of the method for different interval families and for different Jacobi parameters.  \nKeywords Histopolation · Weighted derivative moments · Jacobi polynomials · Chebyshev polynomials  \nMathematics Subject Classification (2020) 41A05  \nAllal Guessab  \nLaboratoire de Math´ematiques et de leurs Applications, UMR CNRS 5142, Universit´e de Pau et des Pays del’Adour (UPPA), France  \nE-mail: [allal.guessab@univ-pau.fr](allal.guessab@univ-pau.fr)  \nFederico Nudo (corresponding author)  \nDepartment of Mathematics and Computer Science, University of Calabria, Rende (CS), Italy E-mail: [federico.nudo@unical.it](federico.nudo@unical.it)  \n1 Introduction  \nApproximation from integral or averaged data is a natural alternative to classical interpolation when pointwise values are not available, or when they do not represent the information provided by the measurement process [9,20] . This situation occurs in several applications in which the data are integrals over intervals, cells, faces, or more general geometric regions, as in tomography and image reconstruction [22,30] . In these cases, the reconstruction procedure should be consistent with the integral nature of the data. This is the basic principle of histopolation, where the degrees of freedom are given by averages or moment type functionals rather than by point evaluations [32] . In this sense, histopolation is a particular case of polynomial interpolation defined by linear functionals. In the classical Lagrange case, these functionals are evaluations at prescribed nodes; in the histopolation case, they are integral averages over prescribed subintervals [","cbCaioSyJ3zT22ij","https://ap.wps.com/l/cbCaioSyJ3zT22ij","pdf",1761091,1,48,"English","en",105,"# Abstract\n# Introduction\n## Weighted histopolation and moment-based data\n## Derivative-moment degrees of freedom\n## Goal: unisolvence, admissibility, and exact factorizations","[{\"question\":\"What are the degrees of freedom in the weighted derivative histopolation scheme?\",\"answer\":\"The scheme uses one scalar normalization together with weighted integral moments of the derivative taken over a prescribed family of subintervals.\"},{\"question\":\"Under what conditions is the interpolation problem unisolvent?\",\"answer\":\"Unisolvence holds when the interval family separates polynomials of degree at most N−1 through weighted moments and the scalar normalization is nonzero on constants.\"},{\"question\":\"How is admissibility determined when interval endpoints lie on a fixed grid?\",\"answer\":\"Admissibility reduces to the nonsingularity of an interval matrix associated with the family, which depends only on the interval representation via consecutive grid cells.\"}]",1784189363,121,{"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},"weighted-derivative-histopolation-on-arbitrary-grids-admissibility-and-exact-factorizations","",{"@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/weighted-derivative-histopolation-on-arbitrary-grids-admissibility-and-exact-factorizations/83626/",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 are the degrees of freedom in the weighted derivative histopolation scheme?","Question",{"text":75,"@type":76},"The scheme uses one scalar normalization together with weighted integral moments of the derivative taken over a prescribed family of subintervals.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Under what conditions is the interpolation problem unisolvent?",{"text":80,"@type":76},"Unisolvence holds when the interval family separates polynomials of degree at most N−1 through weighted moments and the scalar normalization is nonzero on constants.",{"name":82,"@type":73,"acceptedAnswer":83},"How is admissibility determined when interval endpoints lie on a fixed grid?",{"text":84,"@type":76},"Admissibility reduces to the nonsingularity of an interval matrix associated with the family, which depends only on the interval representation via consecutive grid 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