Optimal recovery of isotropic classes of twice-differentiable multivariate functions
The authors consider functions defined on a convex body \(G\subset\mathbb R^d\), \(d\in\mathbb N\), i.e., a compact convex set with non-empty interior, and define two classes of functions. \(W_G\) is the class of continuously differentiable functions \(f:G\rightarrow\mathbb R\), such that the directional derivative of order \(2\) exists inside \(G\) for every direction (at least in a generalized sense) and has an essential supremum on \(G\) bounded by \(1\). \(\widetilde{W}_ {\mathcal L}\) is the class of \(\mathcal L\)-periodic continuous differentiable functions \(f:\mathbb R^d \rightarrow\mathbb R\), such that the directional derivative of order \(2\) exists on \(\mathbb R^d\) for every direction (at least in a generalized sense) and has an essential supremum on the fundamental parallelepiped, bounded by \(1\). Using the so-called central algorithm [c.f., \textit{J. F. Traub} and \textit{H. Wozniakowski}, A general theory of optimal algorithms. ACM Monograph Series. New York etc.: Academic Press (1980; Zbl 0441.68046)], the authors address three problems with respect to optimal recovery operators. The results are stated in 5 theorems (\S3) which are proved in \S\S5--9.
scientific article; zbMATH DE number 5822238
scientific article