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Simultaneous approximation, interpolation and Birkhoff systems. (English) Zbl 0538.41030

Let \(f\in C^{(k_ p)}[a,b]\), \(\| \cdot \|\) be the uniform norm and H the finite dimensional subspace of \(C^{(k_ p)}[a,b]\) with respect to the seminorm \(\| \cdot \|_ F\) given by \(\| \cdot \|_ F=\max_{k\in F}\quad \| f^{(k)}\|,\) where \(F=\{k_ 1,k_ 2,...,k_ p\}\) and \(k_ i\) are integers satisfying \(0\leq k_ 1<k_ 2<...<k_ p\). In this paper various results related to the best simultaneous approximation of a function f by elements of H are established and the precise dimensions of the set \(\Omega\) (f) of the best simultaneous approximations to f determined. In this process the author has improved upon various previous results due to L. L. Keener [ibid. 30, 129-138 (1980; Zbl 0499.41001)].
Reviewer: G.D.Dikshit

MSC:

41A28 Simultaneous approximation
41A05 Interpolation in approximation theory
41A50 Best approximation, Chebyshev systems

Citations:

Zbl 0499.41001
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References:

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