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Optimal interpolation on some class of differentiable functions. (English) Zbl 0639.41003

Given a positive integer \(n\geq 2\). Let \(W_{\infty}^{(n)}({\mathbb{R}})\) be the set of differentiable functions defined on \({\mathbb{R}}\) where for each \(f\in W_{\infty}^{(n)}({\mathbb{R}})f^{(n-1)}\) is absolutely continuous on any interval bounded on \({\mathbb{R}}\) and \(\| f^{(n)}\|_{\infty}=ess \sup | f(x)| \leq 1.\) Denote \(K=W_{\infty}^{(n)}\cap L^{\infty}\). Let \(\Theta\) be the set of all denumerable sets \({\tilde \xi}=\{\xi_ j\}^{+\infty}_{j=-\infty}\) such that for every \({\tilde \xi}\) there is some \(\alpha\), \(2j\leq \xi_ j-\alpha <2(j+1)\) for \(j=0,\pm 1,\pm 2,..\). \({\tilde \Theta}_{2k}\) be the subset of \(\Theta\), where each \({\tilde \xi}\in {\tilde \Theta}_{2k}\) is 2k-periodic, viz., \(\xi_{j+2k=2k+\xi}\xi_ j\) for \(j=0,\pm 1,\pm 2,... \). Denote \({\tilde \Theta}=\cup^{\infty}_{k=1}{\tilde \Theta}_{2k}\) and \(I{\tilde \xi}(f)=\{(f(\xi_ j)),(f'(\xi_ j))\}^{+\infty}_{j=-\infty}.\) Fix a \(\tau\) and consider the problem of recovering optimally the value of \(Uf=f(\tau)\) on the set K by using the information I\({\tilde \xi}\)(f) about \(f\in K\), where \({\tilde \xi}\in {\tilde \Theta}\) is a given denumerable set of sampling. Assuming the information to be error-free, we defined the intrinsic error \[ E(K,{\tilde \xi},\tau)=\inf_{A}\sup_{f\in K}| f(\tau)-AI{\tilde \xi}(f)| \] where A is any mapping (algorithm) from I\({\tilde \xi}\)(K) into \({\mathbb{R}}\). Denoting \(e(K,{\tilde \xi},\tau)=\sup | f(\tau)|,\) \(f\in K\), I\({\tilde \xi}\)(f)\(=0\), we have \(E(K,{\tilde \xi},\tau)\geq e(K,{\tilde \xi},\tau).\) Now we put \(E(K,{\tilde \xi})=\| E(K,{\tilde \xi},\cdot)\|_{\infty}\) and \(E(K)=\inf \{E(K,{\tilde \xi}):{\tilde \xi}\in {\tilde \Theta}\}.\) In this note our objective is to obtain the exact expression of E(K) and an optimal set of sampling points \({\tilde \xi}_ 0\in {\tilde \Theta}\), viz. a \({\tilde \xi}_ 0\in {\tilde \Theta}\) for which \(E(K)=E(K,{\tilde \xi}_ 0)\). Theorem: Let \(n\geq 2\), \({\tilde \Theta}\), I\({\tilde \xi}\)(f) and K be given as above. Then we have \(E(K)=\| E_ n(x);\| E_ n(\cdot)\|_{\infty}\|_{\infty},\) where \(E_ n(x)\) is the Euler spline of degree n.
Reviewer: Yongsheng Sun

MSC:

41A05 Interpolation in approximation theory
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