\(L^{1}\)-approximation and finding solutions with small support (Q1935027)

As noted in the abstract, ``for many subspaces \(M\), there exist \(\alpha^*(M)>0\) with the following property: if \(f\) vanishes off a set of measure at most \(\alpha^*(M)\), then the zero function is a best \(L^1\)-approximant to \(f\) from \(M\).'' The authors explain this phenomenon, starting with the characterization of the best approximation from a linear subspace in the \(L^1\)-norm. Several specific examples of subspaces, with lower bounds and sometimes upper bounds on the associated \(\alpha^*(M)\) are given. Finally, the authors present some open questions.