A sharp upper bound on the approximation order of smooth bivariate pp functions
Let \(\Pi_{k,\Delta}^ \rho\) denote the set of all piecewise polynomial functions in \(C^ \rho\) on a triangulation \(\Delta\) of \(\mathbb{R}^ 2\), of polynomial degree not exceeding \(k\). The approximation order of a space \(S\) of functions is defined as the largest number \(r\) for which \(\text{dist}(f,\sigma_ h S)_ p \leq \text{const}_ f h^ r\) for every sufficiently smooth \(f\) in \(L_ p\), \(1\leq p\leq\infty\), of \(\mathbb{R}^ 2\) or of some suitable subset of \(\mathbb{R}^ 2\). The scaling map \(\sigma_ h\) is defined as usual, by \(\sigma_ h f:= f(\cdot/h)\). It has been known [\textit{C. de Boor} and \textit{K. Höllig}, Math. Z. 197, No. 3, 343-363 (1988; Zbl 0616.41010)] that if \(k\geq 3\rho+2\), then the approximation of order \(\Pi_{k,\Delta}^ \rho\) is the best possible, namely \(k+1\), for an arbitrary triangulation \(\Delta\). In case \(\rho=-1\) or 0, then again, it is trivially \(k+1\). This paper is devoted to proving that for \(\rho>0\) the approximation order of \(\Pi_{k,\Delta}^ \rho\) is, in general no better than \(k\) when \(k<3\rho+2\). Thus complementing the above mentioned results.
scientific article; zbMATH DE number 222113
scientific article