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On minimal and quasi-minimal supported bivariate splines. (English) Zbl 0644.41007
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.
Reviewer: L.Leindler

##### MSC:
 41A15 Spline approximation
##### Keywords:
bivariate splines
Full Text:
##### References:
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