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On an $L_ 1$-approximation problem


Author: András Kroó
Journal: Proc. Amer. Math. Soc. 94 (1985), 406-410
MSC: Primary 41A50; Secondary 41A52
DOI: https://doi.org/10.1090/S0002-9939-1985-0787882-0
MathSciNet review: 787882
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Abstract: Let ${C_w}[a,b]$ denote the space of real continuous functions with norm ${\left \| f \right \|_w} = \smallint _a^b\left | {f(x)} \right |w(x)dx$, 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