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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the strong unicity of best Chebyshev approximation of differentiable functions


Author: András Kroó
Journal: Proc. Amer. Math. Soc. 89 (1983), 611-617
MSC: Primary 41A52
DOI: https://doi.org/10.1090/S0002-9939-1983-0718983-9
MathSciNet review: 718983
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Abstract: Let $X$ be a normed linear space, ${U_n}$ an $n$-dimensional Chebyshev subspace of $X$. For $f \in X$ denote by $p(f) \in {U_n}$ its best approximation in ${U_n}$. The problem of strong unicity consists in estimating how fast the nearly best approximants $g \in {U_n}$ satisfying $\left \| {f - g} \right \| \leqslant \left \| {f - p(f)} \right \| + \delta$ approach $p(f)$ as $\delta \to 0$. In the present note we study this problem in the case when $X = {C^r}[a,b]$ is the space of $r$-times continuously differentiable functions endowed with the supremum norm $(1 \leqslant r \leqslant \infty )$.


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Article copyright: © Copyright 1983 American Mathematical Society