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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)
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