Stieltjes polynomials and Lagrange interpolation

Abstract

Bounds are proved for the Stieltjes polynomial $E_{n+1}$, and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials $P_n$. This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials $G_n$. Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of $E_{n+1}$, and for the extended Lagrange interpolation process with respect to the zeros of $P_n E_{n+1}$ in the uniform and weighted $L^p$ norms. The corresponding Lebesgue constants are of optimal order.