A. Romanyuk, V. Romanyuk, K. Pozharska, S. Hembars’ka
{"title":"周期多元函数各向同性类的线性和非线性逼近的特征","authors":"A. Romanyuk, V. Romanyuk, K. Pozharska, S. Hembars’ka","doi":"10.15330/cmp.15.1.78-94","DOIUrl":null,"url":null,"abstract":"Exact order estimates for some characteristics of linear and nonlinear approximation of the isotropic Nikol'skii-Besov classes $\\mathbf{B}^r_{p,\\theta}$ of periodic multivariate functions in the spaces $B_{q,1}$, $1\\leq q \\leq \\infty$, are obtained. Among them are the best orthogonal trigonometric approximations, best $m$-term trigonometric approximations, Kolmogorov, linear and trigonometric widths. \nFor all considered characteristics, their estimates coincide in order with the corresponding estimates in the spaces $L_{q}$. Moreover, the obtained exact in order estimates (except the case $1<p<2\\leq q < \\frac{p}{p-1}$) are realized by the approximation of functions from the classes ${\\mathbf{B}}^r_{p,\\theta}$ by trigonometric polynomials with the spectrum in cubic regions. In any case, they do not depend on the smoothness parameter $\\theta$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Characteristics of linear and nonlinear approximation of isotropic classes of periodic multivariate functions\",\"authors\":\"A. Romanyuk, V. Romanyuk, K. Pozharska, S. Hembars’ka\",\"doi\":\"10.15330/cmp.15.1.78-94\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Exact order estimates for some characteristics of linear and nonlinear approximation of the isotropic Nikol'skii-Besov classes $\\\\mathbf{B}^r_{p,\\\\theta}$ of periodic multivariate functions in the spaces $B_{q,1}$, $1\\\\leq q \\\\leq \\\\infty$, are obtained. Among them are the best orthogonal trigonometric approximations, best $m$-term trigonometric approximations, Kolmogorov, linear and trigonometric widths. \\nFor all considered characteristics, their estimates coincide in order with the corresponding estimates in the spaces $L_{q}$. Moreover, the obtained exact in order estimates (except the case $1<p<2\\\\leq q < \\\\frac{p}{p-1}$) are realized by the approximation of functions from the classes ${\\\\mathbf{B}}^r_{p,\\\\theta}$ by trigonometric polynomials with the spectrum in cubic regions. In any case, they do not depend on the smoothness parameter $\\\\theta$.\",\"PeriodicalId\":42912,\"journal\":{\"name\":\"Carpathian Mathematical Publications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-06-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Carpathian Mathematical Publications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.15330/cmp.15.1.78-94\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Carpathian Mathematical Publications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.15330/cmp.15.1.78-94","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Characteristics of linear and nonlinear approximation of isotropic classes of periodic multivariate functions
Exact order estimates for some characteristics of linear and nonlinear approximation of the isotropic Nikol'skii-Besov classes $\mathbf{B}^r_{p,\theta}$ of periodic multivariate functions in the spaces $B_{q,1}$, $1\leq q \leq \infty$, are obtained. Among them are the best orthogonal trigonometric approximations, best $m$-term trigonometric approximations, Kolmogorov, linear and trigonometric widths.
For all considered characteristics, their estimates coincide in order with the corresponding estimates in the spaces $L_{q}$. Moreover, the obtained exact in order estimates (except the case $1
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