Best one-sided \(L_1\) approximation to the Heaviside and sign functions (Q435180)

An explicit expression for polynomials of best one-sided approximation to the Heaviside function~\(G\) (and hence to the sign function) is obtained. Characterization of the polynomials in \(\mathcal P^{-1}(G)\) is established. The polynomials are obtained by Hermite interpolation at the zeros of some Jacobi polynomials. An estimate of the error of approximation is given, and the extreme points of the convex set of the best approximants are characterized.