On simultaneous best \(L_ 1\) approximations in C[-1,1] (Q789634)

This paper considers the same problem posed in the paper reviewed above with the \(L_ 1\) norm and gives the following result: Let \(V_ 1\) and \(V_ 2\) be Chebyshev subspaces of C[-1,1] with dimensions 1 and \(n(n>1)\), respectively. Let \(V_ 1\subset V_ 2\) and \(v_ j\in V_ j (j=1,2)\). Then there exists an \(f\in C[-1,1]\) such that \(v_ j\) is a best \(L_ 1\) approximation to f from \(V_ j (j=1,2)\) iff \(v=v_ 2-v_ 1\) changes sign at least once in [-1,1] or is equal to zero.