{"title":"统一拓扑中的马查多-毕夏普定理","authors":"Deliang Chen","doi":"10.1016/j.jat.2024.106085","DOIUrl":null,"url":null,"abstract":"<div><p>The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>×</mo><mi>J</mi><mo>,</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span> as the closure of <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow><mo>⊗</mo><mi>C</mi><mrow><mo>(</mo><mi>J</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span> in different senses and the extension of continuous vector-valued functions are studied.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Machado–Bishop theorem in the uniform topology\",\"authors\":\"Deliang Chen\",\"doi\":\"10.1016/j.jat.2024.106085\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>×</mo><mi>J</mi><mo>,</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span> as the closure of <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo>)</mo></mrow><mo>⊗</mo><mi>C</mi><mrow><mo>(</mo><mi>J</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span> in different senses and the extension of continuous vector-valued functions are studied.</p></div>\",\"PeriodicalId\":54878,\"journal\":{\"name\":\"Journal of Approximation Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-07-26\",\"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/S002190452400073X\",\"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/S002190452400073X","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The Machado–Bishop theorem in the uniform topology
The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado’s distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of as the closure of in different senses and the extension of continuous vector-valued functions are studied.
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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