On the Lebesgue function of weighted Lagrange interpolation.: I.: (Freud-type weights)

For a wide class of Freud-type weights of form w = exp(-Q) we investigate the behavior of the corresponding weighted Lebesgue function lambda(n)(w, X, x), where X = {x(kn)} subset of (-infinity, infinity) is an interpolatory matrix. We prove that for arbitrary X subset of (-infinity, infinity) and epsilon > 0, lambda(n)(w, X, x) greater than or equal to c epsilon log n, x is an element of [-a(n), a(n)]\H-n, n greater than or equal to 1, where a(n) is the MRS number and \H-n\ less than or equal to 2 epsilon a(n). The result corresponds to the behavior of the "ordinary" Lebesgue function in [-1, 1]. Other exponential weights are considered in our subsequent paper.