The Chebyshev rank of multivariate polynomials in \(L^1\)-approximation (Q1958432)

Let \(C\) be the continuous functions defined on the closure of a bounded open subset of \(\mathbb R^d\). For a subset \(M\) of \(C\), and an \(f\) in \(C\), let \(P_Mf\) be the set of best \(L_1\)-approximations to \(f\) from \(M\),where the measure is a weighted Lebesgue measure. This work determines the dimension of the affine space spanned by \(P_Mf\) when \(M\) is the important class of multivariate polynomials of a specific finite degree. The results focus on a condition (called property \(A^k\)) that is a generalization of a property used to study unique best \(L_1\)-approximations on the line.