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Upper and Lower Bounds on the Dimension of Multivariate Spline Spacesy Peter Alfeld*
 

Summary: Upper and Lower Bounds on the Dimension of Multivariate Spline Spacesy
by
Peter Alfeld*
List of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. The Bernstein B'ezier form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3. Two Simple Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4. A Simple Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5. A Refined Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6. Comparison of Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Abstract
We give upper and lower bounds on the dimensions of multivariate spline
spaces defined on triangulations in IR k . The bounds are optimal in a certain
sense.
1. Introduction
In this paper we consider spaces of splines (smooth piecewise polynomial

  

Source: Alfeld, Peter - Department of Mathematics, University of Utah

 

Collections: Mathematics