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On bivariate super vertex splines. (English) Zbl 0726.41012
The authors construct a vertex spline basis for the super spline subspace $$\hat S^ r_ d(\Delta)$$ of $$S^ r_ d(\Delta)$$. Here $$\Delta$$ means an arbitrary regular triangulation in $${\mathbb{R}}^ 2$$, $S_ d^{r,\ell}(\Delta)=\{s\in S^ r_ d(\Delta),\;D^{\alpha}s(v)\text{ exists for } | \alpha | \leq \ell \text{ and every vertex $$v$$ of }\Delta\}$ and a vertex spline of $$\hat S^ r_ d(\Delta)=S_ d^{r,r+\lfloor (d-2r-1)/2\rfloor}(\Delta)$$ has a support which contains at most one vertex of $$\Delta$$ in its interior. C. de Boor and K. Höllig [Math. Z. 197, 343-363 (1987; Zbl 0616.41010)] proved that $$S^ r_ d(\Delta)$$ has approximation order $$d+1$$ provided that $$d\geq 3r+2$$. Here it is shown that this can be achieved already by using avertex spline basis. Therefore a quasi-interpolatory linear operator L is considered which reproduces functions from $$\hat S^ r_{3r+2}(\Delta)$$. Then for $$d\geq 3r+2$$ and sufficiently smooth functions f it holds $$\| f-Lf\| \leq C\| D^{d+1}f\| | \Delta |^{d+1}.$$ The proofs are based on the Bernstein-Bézier technique, which gives a very constructive approach to the vertex splines.

##### MSC:
 41A15 Spline approximation 41A25 Rate of convergence, degree of approximation 41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
##### Keywords:
super spline; Bernstein-Bézier technique; vertex splines
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##### References:
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