{"title":"Log-concavity of B-splines","authors":"Michael S. Floater","doi":"10.1016/j.jat.2024.106042","DOIUrl":null,"url":null,"abstract":"<div><p>Curry and Schoenberg showed that a B-spline is log-concave in its support by applying Brunn’s theorem to a simplex. In this note we provide an alternative, ‘analytic’ proof of the log-concave property using only recursion formulas for B-splines and their first and second derivatives.</p></div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021904524000285/pdfft?md5=e7cbce1cee37c3e76009eab70b3d59a1&pid=1-s2.0-S0021904524000285-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Approximation Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021904524000285","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Curry and Schoenberg showed that a B-spline is log-concave in its support by applying Brunn’s theorem to a simplex. In this note we provide an alternative, ‘analytic’ proof of the log-concave property using only recursion formulas for B-splines and their first and second derivatives.
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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