Book chapter
On a New Proximity Condition for Manifold-Valued Subdivision Schemes
Approximation Theory XIV: San Antonio 2013, pp 65-79
03 Jun 2014
Abstract
An open theoretical problem in the study of subdivision algorithms for approximation of manifold-valued data has been to give necessary and sufficient conditions for a manifold-valued subdivision scheme, based on a linear subdivision scheme, to share the same regularity as the linear scheme. This is called the smoothness equivalence problem. In a companion paper, the authors introduced a differential proximity condition that solves the smoothness equivalence problem. In this paper, we review this condition, comment on a few of its unanticipated features, and as an application, show that the single basepoint log-exp scheme suffers from an intricate breakdown of smoothness equivalence. We also show that the differential proximity condition is coordinate independent, even when the linear scheme is not assumed to possess the relevant smoothness.
Metrics
8 Record Views
3 citations in Scopus
Details
- Title
- On a New Proximity Condition for Manifold-Valued Subdivision Schemes
- Creators
- Tom Duchamp - University of WashingtonGang Xie - East China University of Science and TechnologyThomas Yu - Drexel University
- Publication Details
- Approximation Theory XIV: San Antonio 2013, pp 65-79
- Series
- Springer Proceedings in Mathematics & Statistics
- Publisher
- Springer International Publishing; Cham
- Resource Type
- Book chapter
- Language
- English
- Academic Unit
- Mathematics
- Scopus ID
- 2-s2.0-84927671016
- Other Identifier
- 991021878114204721