{"title":"齐次混合范数triiebel - lizorkin空间的稳定分解","authors":"Morten Nielsen","doi":"10.1016/j.jat.2023.105958","DOIUrl":null,"url":null,"abstract":"<div><p>We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel–Lizorkin spaces.</p><p>In the second part of the paper we study nonlinear <span><math><mi>m</mi></math></span>-term approximation with the constructed basis in the mixed-norm setting, where the behavior, in general, for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for <span><math><mi>m</mi></math></span>-term approximation can still be derived.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Stable decomposition of homogeneous Mixed-norm Triebel–Lizorkin spaces\",\"authors\":\"Morten Nielsen\",\"doi\":\"10.1016/j.jat.2023.105958\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel–Lizorkin spaces.</p><p>In the second part of the paper we study nonlinear <span><math><mi>m</mi></math></span>-term approximation with the constructed basis in the mixed-norm setting, where the behavior, in general, for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for <span><math><mi>m</mi></math></span>-term approximation can still be derived.</p></div>\",\"PeriodicalId\":54878,\"journal\":{\"name\":\"Journal of Approximation Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-08-22\",\"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/S0021904523000965\",\"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/S0021904523000965","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Stable decomposition of homogeneous Mixed-norm Triebel–Lizorkin spaces
We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on . The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel–Lizorkin spaces.
In the second part of the paper we study nonlinear -term approximation with the constructed basis in the mixed-norm setting, where the behavior, in general, for , is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for -term approximation can still be derived.
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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