On an 𝐿₁-approximation problem

Abstract

Let C w [ a , b ] {C_w}[a,b] denote the space of real continuous functions with norm ‖ f ‖ w = ∫ a b | f ( x ) | w ( x ) d x {\left \| f \right \|_w} = \smallint _a^b\left | {f(x)} \right |w(x)dx , where w w is a positive bounded weight. It is known that if a subspace M n ⊂ C w [ a , b ] {M_n} \subset {C_w}[a,b] satisfies a certain A A -property, then M n {M_n} is a Chebyshev subspace of C w [ a , b ] {C_w}[a,b] for all w w . We prove that the A A -property is also necessary for M n {M_n} to be Chebyshev in C w [ a , b ] {C_w}[a,b] for each w w .