Saturation problem of \(L^ p\)-approximation by Hermite-Fejér interpolation (Q1125967)

The saturation of \(L^p\)-approximation of Hermite-Fejér interpolation based on the zeros of generalized Jacobi polynomials is considered. Although mean convergence may improve the approximation order compared to uniform convergence, surprisingly, their saturation orders are exactly the same, that is, \(1/n\). An inverse theorem is also given with respect to \(L^p\)-approximation of Hermite-Fejér interpolation.