Best 𝐿₁-approximation with varying weights

Abstract

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