Best $L_ 1$-approximation with varying weights

Abstract:

It is proved in this note that the so-called $A$-property is necessary in order that the finite-dimensional space $U$ be Chebyshev in $C\left ( K \right )$ with respect to the norm $\left \| f \right \| = \int _K {\omega \left | f \right |}$ for every positive continuous weight $\omega$. It is also shown that for each finite-dimensional subspace $U$ there exists a positive continuous weight $\omega$ such that $U$ is Chebyshev in $C\left ( K \right )$ with respect to this weight $\omega$.