NDLI: Convergence and Smoothness of Nonlinear Lane–Riesenfeld Algorithms in the Functional Setting

Content Provider	Springer Nature Link
Author	Dyn, Nira Goldman, Ron
Copyright Year	2010
Abstract	We investigate the Lane–Riesenfeld subdivision algorithm for uniform B-splines, when the arithmetic mean in the various steps of the algorithm is replaced by nonlinear, symmetric, binary averaging rules. The averaging rules may be different in different steps of the algorithm. We review the notion of a symmetric binary averaging rule, and we derive some of its relevant properties. We then provide sufficient conditions on the nonlinear binary averaging rules used in the Lane–Riesenfeld algorithm that ensure the convergence of the algorithm to a continuous function. We also show that, when the averaging rules are C 2 with uniformly bounded second derivatives, then the limit is a C 1 function. A canonical family of nonlinear, symmetric averaging rules, the p-averages, is presented, and the Lane–Riesenfeld algorithm with these averages is investigated.
Ending Page	94
Page Count	16
Starting Page	79
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	1
Volume Number	11
Language	English
Publisher	Springer-Verlag
Publisher Date	2010-11-10
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Computer aided design (modeling of curves and surfaces) Economics general Splines Data refinement Linear and Multilinear Algebras, Matrix Theory Convergence and smoothness analysis Nonlinear subdivision schemes Spline approximation Lane–Riesenfeld algorithm p-averages Numerical Analysis Computer Science Nonlinear symmetric averages Math Applications in Computer Science Applications of Mathematics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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