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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weak Chebyshev subspaces and continuous selections for the metric projection

Authors: Günther Nürnberger and Manfred Sommer
Journal: Trans. Amer. Math. Soc. 238 (1978), 129-138
MSC: Primary 41A50; Secondary 41A65
MathSciNet review: 482912
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Abstract: Let G be an n-dimensional subspace of $ C[a,b]$. It is shown that there exists a continuous selection for the metric projection if for each f in $ C[a,b]$ there exists exactly one alternation element $ {g_f}$, i.e., a best approximation for f such that for some $ a \leqslant {x_0} < \cdots < {x_n} \leqslant b$,

$\displaystyle \varepsilon {( - 1)^i}(f - {g_f})({x_i}) = \left\Vert {f - {g_f}} \right\Vert,\quad i = 0, \ldots ,n,\varepsilon = \pm 1.$

Further it is shown that this condition is fulfilled if and only if G is a weak Chebyshev subspace with the property that each g in G, $ g \ne 0$, has at most n distinct zeros. These results generalize in a certain sense results of Lazar, Morris and Wulbert for $ n = 1$ and Brown for $ n = 5$.

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Keywords: Continuous selection, metric projection, weak Chebyshev spaces, alternation elements
Article copyright: © Copyright 1978 American Mathematical Society