Stieltjes polynomials and Lagrange interpolation

Bounds are proved for the Stieltjes polynomial En+1, and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials Pn. This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials Gn. Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of En+1, and for the extended Lagrange interpolation process with respect to the zeros of PnEn+1 in the uniform and weighted Lp norms. The corresponding Lebesgue constants are of optimal order.