Algebraic methods in approximation theory

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• 作者：Hal Schenck¹ ; ^{schenck@math.uiuc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Spline ; Polyhedral complex ; Homology ; Localization ; Inverse system
• 刊名：Computer Aided Geometric Design
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：45
• 期：Complete
• 页码：14-31
• 全文大小：466 K

文摘

This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis; their properties depend on combinatorics, topology, and geometry of a simplicial or polyhedral subdivision of a region in R^k${\mathbb{R}}^k$, and are often quite subtle. We describe four algebraic techniques which are useful in the study of splines: homology, graded algebra, localization, and inverse systems. Our goal is to give a hands-on introduction to the methods, and illustrate them with concrete examples in the context of splines. We highlight progress made with these methods, such as a formula for the third coefficient of the polynomial giving the dimension of the spline space in high degree. The objects appearing here may be computed using the spline package of the Macaulay2 software system.