On an $L_ 1$-approximation problem

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 $w$.