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On an $ L\sb 1$-approximation problem

Author: András Kroó
Journal: Proc. Amer. Math. Soc. 94 (1985), 406-410
MSC: Primary 41A50; Secondary 41A52
MathSciNet review: 787882
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Abstract: Let $ {C_w}[a,b]$ denote the space of real continuous functions with norm $ {\left\Vert f \right\Vert _w} = \smallint _a^b\left\vert {f(x)} \right\vert w(x)dx$, where $ w$ is a positive bounded weight. It is known that if a subspace $ {M_n} \subset {C_w}[a,b]$ satisfies a certain $ A$-property, then $ {M_n}$ is a Chebyshev subspace of $ {C_w}[a,b]$ for all $ w$. We prove that the $ A$-property is also necessary for $ {M_n}$ to be Chebyshev in $ {C_w}[a,b]$ for each $ w$.

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Article copyright: © Copyright 1985 American Mathematical Society