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Optimal recovery of isotropic classes of twice-differentiable multivariate functions. (English) Zbl 1215.41001

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., J. F. Traub and 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 (§3) which are proved in §§5–9.

MSC:

41A05 Interpolation in approximation theory
41A29 Approximation with constraints
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)

Citations:

Zbl 0441.68046

Software:

kepler98
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References:

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