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Proceedings of the American Mathematical Society

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

 

 

Bounds for nearly best approximations


Author: Rudolf Wegmann
Journal: Proc. Amer. Math. Soc. 52 (1975), 252-256
MSC: Primary 41A65
DOI: https://doi.org/10.1090/S0002-9939-1975-0442563-1
MathSciNet review: 0442563
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Abstract: Let $ X$ be a uniformly convex space and $ \psi $ be the inverse function of the modulus of convexity $ \delta ( \cdot )$. Assume here that $ \psi $ is a concave function. Let $ V$ be a linear subspace of $ X$ and let $ f$ in $ X$ be such that $ \vert\vert f\vert\vert = 1 = \min \{ \vert\vert f - v\vert\vert:v\epsilon V\} $. Then for $ 0 < \delta \leqslant 1$ and for $ v$ in $ V$ with $ \vert\vert f - v\vert\vert \leqslant 1 + \delta $, it follows that $ \vert\vert v\vert\vert \leqslant K \cdot \psi (\delta )$. Let $ T$ be a compact Hausdorff-space and $ V$ a finite-dimensional subspace of $ C(T,X)$. When $ V$ has the interpolation property $ ({P_m})$ with $ V = m \cdot \dim X$, then the same type of estimate as above holds.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0442563-1
Article copyright: © Copyright 1975 American Mathematical Society