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Approximation order from bivariate $$C^ 1$$-cubics: A counterexample. (English) Zbl 0545.41017
A fixed space S and $$S_ h:=\sigma_ h(S)$$ with $$(\sigma_ hf)(x):=f(x/h)$$, and ”Condition $$P_ k''$$ which allows the polynomials $$P_ k$$ of degree $$<k$$ to be contained in S ”locally” are given. In section 2 bivariate piecewise polynomial (pp) functions on the partition $$\Delta$$ of $${\mathbb{R}}^ 2$$ defined by $$x(1)=n$$, $$x(2)=n$$, $$x(1)=x(2)+n,$$ $$n\in {\mathbb{Z}}$$, and the space $$S:=P^ 1_{4,\Delta}:=P_{4,\Delta}\cap C^ 1({\mathbb{R}}^ 2)$$ of piecewise cubic functions on $$\Delta$$ and in $$C^ 1$$ are considered. It is shown that S satisfies condition $$P_ 4$$ and that $$(S_ h)$$ has approximation order $$O(h^ 3)$$ at least.
In section 3 S is identified as a subspace of $$P^ 0_{4,\Delta}$$ satisfying certain homogeneous conditions which are expressed in terms of the Bernstein coordinates for pp functions on a triangulation. S is characterized as the annihilator of a set $$\Lambda$$ of local linear functionals. Furthermore, it is shown that there exists a bounded $$\alpha \in {\mathbb{R}}^{\Lambda}\backslash 0$$ such that $$\sum_{\lambda \in \Lambda}\alpha(\lambda)\lambda f=0$$ for all f with compact support, while $$\sum_{\sup p \Lambda \leq G}\alpha(\lambda)\lambda =0$$ only if the sum is over all or over none of $$\Lambda$$. This yields the basis for section 5 where it is proved that the approximation order of $$(S_ h)$$ is $$O(h^ 3)$$ at most.
Reviewer: R.Fahrion

##### MSC:
 41A15 Spline approximation 41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX) 41A25 Rate of convergence, degree of approximation
##### Keywords:
B-splines; piecewise cubic functions
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