Quasi orthogonal Jacobi polynomials and best one-sided L1L1 approximation to step functions

We find the polynomials of the best one-sided approximation to a step function on [−1,1][−1,1]. We prove that these polynomials are obtained by Hermite interpolation at the zeros of some quasi orthogonal Jacobi polynomials. We discuss the cases when there is uniqueness and, if there is not, we determine the extremal points of the convex set of the best approximants.