Mean convergence of interpolation polynomials for exponential weights

Abstract

The purpose of this work is to study mean and uniform convergence of Hermite, Hermite-Fej&eaute;r and Lagrange interpolation polynomials based at zeros of orthogonal polynomial with respect to Freud weights, ErdÖs weights on R and Exponential weights on (-1,1). D. S. Lubinsky, D. Matjila and S. B. Damelin obtained results about necessary and sufficient conditions for $L_p(1＜p＜∞)$ convergence of Lagrange interpolation based at the zeros of orthogonal polynomials for the Freud, ErdÖs, and Exponential weights on (-1,1), respectively. From J. Szabados`` observation by adding two points at the set of zeros, D. S. Lubinsky and G. Mastroianni improved Lubnisky and Matjila``s results for the Freud weights and D. S. Lubinsky obtained improved results for the exponential weights on (-1,1). On the other hand, J. Szabados studied uniform convergence of Lagrange interpolation for Freud weights with respect to the zeros of orthogonal polynomials together with two additional points. This choice of points was shown to yield more optimal $L_∞$ results than those of D. Matjila. S. B. Damelin generalized the results of J. Szabados to ErdÖs, weights and non SzegÖ class weights on (-1,1). In Chapter 3, we obtain necessary and sufficient conditions for $L_p(0＜p≤1)$ convergence of Lagrange interpolation and we improve results for the case of ErdÖs weights by estimates of converse Marcinkiewicz-Zygmund inequalities in adding two points. Also, we establish a necessary condition for weighted $L_p(0＜p＜∞)$ convergence of Lagrange interpolation for continuous functions f which vanish outside finite fixed interval J ⊂ R or (-1,1) in case of Freud, ErdÖs weights as well as exponential weights on (-1,1). Our class of functions is the smallest for which convergence questions in weighted $L_{p}$ spaces are meaningful and so our necessary condition is the least we can expect to achieve convergence. D. S. Lubinsky and P. Rabinowitz, motivated by earlier work of P. ...