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Dimension and Local Bases of Homogeneous Spline Spaces

Summary: Dimension and Local Bases of
Homogeneous Spline Spaces
Peter Alfeld 1) , Marian Neamtu 2) , and Larry L. Schumaker 3)
Abstract. Recently, we have introduced spaces of splines defined on trian­
gulations lying on the sphere or on sphere­like surfaces. These spaces arose
out of a new kind of Bernstein­B'ezier theory on such surfaces. The purpose
of this paper is to contribute to the development of a constructive theory for
such spline spaces analogous to the well­known theory of polynomial splines
on planar triangulations. Rather than working with splines on sphere­like
surfaces directly, we instead investigate more general spaces of homogeneous
splines in IR 3 . In particular, we present formulae for the dimensions of such
spline spaces, and construct locally supported bases for them.
1. Introduction
Let \Delta := fT [i] g N
1 be a planar triangulation of a
set\Omega\Gamma and let 0 Ÿ r Ÿ d be
integers. The classical space of splines of degree d and smoothness r is defined
S r


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


Collections: Mathematics