{"title":"基于核的空间的H-集","authors":"Robert Schaback","doi":"10.1016/j.jat.2023.105942","DOIUrl":null,"url":null,"abstract":"<div><p>The concept of <span><math><mi>H</mi></math></span><span>-sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and </span><span><math><mi>H</mi></math></span><span>-sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, </span><span><math><mi>H</mi></math></span>-sets are shown here to have a rather simple and complete characterization. As a byproduct, the strong connection of <span><math><mi>H</mi></math></span>-sets to Linear Programming is studied. But on the downside, it is explained why <span><math><mi>H</mi></math></span>-sets have a very limited range of applicability in the times of large-scale computing.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"H-sets for kernel-based spaces\",\"authors\":\"Robert Schaback\",\"doi\":\"10.1016/j.jat.2023.105942\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The concept of <span><math><mi>H</mi></math></span><span>-sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and </span><span><math><mi>H</mi></math></span><span>-sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, </span><span><math><mi>H</mi></math></span>-sets are shown here to have a rather simple and complete characterization. As a byproduct, the strong connection of <span><math><mi>H</mi></math></span>-sets to Linear Programming is studied. But on the downside, it is explained why <span><math><mi>H</mi></math></span>-sets have a very limited range of applicability in the times of large-scale computing.</p></div>\",\"PeriodicalId\":54878,\"journal\":{\"name\":\"Journal of Approximation Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Approximation Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021904523000801\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Approximation Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021904523000801","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The concept of -sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and -sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, -sets are shown here to have a rather simple and complete characterization. As a byproduct, the strong connection of -sets to Linear Programming is studied. But on the downside, it is explained why -sets have a very limited range of applicability in the times of large-scale computing.
期刊介绍：
The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others:
• Classical approximation
• Abstract approximation
• Constructive approximation
• Degree of approximation
• Fourier expansions
• Interpolation of operators
• General orthogonal systems
• Interpolation and quadratures
• Multivariate approximation
• Orthogonal polynomials
• Padé approximation
• Rational approximation
• Spline functions of one and several variables
• Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds
• Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth)
• Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis
• Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth)
• Gabor (Weyl-Heisenberg) expansions and sampling theory.
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