On minimal and quasi-minimal supported bivariate splines
type: Item
instance of: scholarly article
title: On minimal and quasi-minimal supported bivariate splines
zbMATH Open document ID: 0644.41007
DOI: 10.1016/0021-9045(88)90057-3
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: Let \(\Delta\) be a grid partition of the plane \({\mathbb{R}}^ 2\) with grid lines \(x=i\), \(y=i\), \(x+y=i\), and \(x-y=i\), where \(i=...,-1,0,1,..\). \(S^ k_ d(\Delta)\) denotes the space of all functions in \(C^ k({\mathbb{R}}^ 2)\) whose restriction to each triangular cell of the partition \(\Delta\) is a restriction of an element \(\pi_ d\), the space of all polynomials in x and y of total degree at most d. \(S^ k_ d(\Delta)\) is called a space of bivariate splines. The authors construct two minimal supported (ms) bivariate splines in \(S_ r=S^{3r+2}_{4r+4}(\Delta)\) and prove, surprisingly, that every ms function in this space with convex support is some constant multiple of a translate of one of them. Some further interesting results are proved; e.g. they also construct a quasiminimal supported (qms) bivariate spline in \(S_ r\), and prove that any qms function in this space with convex support is some constant multiple of a translate of this function modulo an ms function.
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 4050197
zbMATH Keywords: bivariate splines
author: Charles K. Chui
author: Tian-Xiao He
reviewed by: László Leindler
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90057-3
OpenAlex ID: W1976956088
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description: scientific article; zbMATH DE number 4050197