Mean convergence of extended Lagrange interpolation for exponential weights

Abstract

In this paper, we complete our investigations of mean convergence of Lagrange interpolation for fast decaying even and smooth exponential weights on the line. In doing so, we also present a summary of recent related work on the line and [-1, 1] by the authors, Szabados, Vertesi, Lubinsky and Matjila. We also emphasize the important and fundamental ideas, applied in our proofs, that were developed by Erdos, Turan, Askey, Freud, Nevai, Szabados, Vertesi and their students and collaborators. These methods include forward quadrature estimates, orthogonal expansions, Hilbert transforms, bounds on Lebesgue functions and the uniform boundedness principle.