Improving the convergence analysis of linear subdivision schemes
Abstract: This work presents several new results concerning the analysis of the convergence of binary, univariate, and linear subdivision schemes, all related to the {\it contractivity factor} of a convergent scheme. First, we prove that a convergent scheme cannot have a contractivity factor lower than half. Since the lower this factor is, the faster is the convergence of the scheme, schemes with contractivity factor $\frac{1}{2}$, such as those generating spline functions, have optimal convergence rate. Additionally, we provide further insights and conditions for the convergence of linear schemes and demonstrate their applicability in an improved algorithm for determining the convergence of such subdivision schemes.
- Nira Dyn. Subdivision schemes in computer-aided geometric design. In Advances in Numerical Analysis: Wavelets, Subdivision Algorithms, and Radial Basis Functions, Will Light (ed.), pages 36–104. Oxford University Press, 1992.
- Nira Dyn. Analysis of convergence and smoothness by the formalism of laurent polynomials. Tutorials on Multiresolution in Geometric Modelling: Summer School Lecture Notes, pages 51–68, 2002.
- Analysis of Hermite-type subdivision schemes. Series in Approximations and Decompositions, 6:117–124, 1995.
- Avraham A Melkman. Subdivision schemes with non-negative masks converge always-unless they obviously cannot? Annals of Numerical Mathematics, 4:451–460, 1996.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.