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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.

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)
Full Text: DOI
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