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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

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
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