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
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\Vert {f - g} \right\Vert \leqslant \left\Vert {f - p(f)} \right\Vert + \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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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0718983-9
Article copyright: © Copyright 1983 American Mathematical Society