The general question asks when there is a function that has a prescribed set of approximates one from each set in a nested family. The characterization here is for the important class of \(L_ 1\) approximation from Chebyshev spaces. Let \(V_ n\) be a Chebyshev subspace of dimension n and assume \(V_ m\subseteq V_ n\) for \(m\leq n\). Let \(v_ i\in V_ i\). Fix two indices m and n and let \(v=v_ n-v_ m\). Theorem: (a) If v has m sign changes there is an f such that \(v_ i\) is the best approximation to f from \(V_ i (i=m,n)\). (b) If v has \(v_ i\) as a best approximation from \(V_ i (i=m,n)\) than v has m zeros.