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 λ _n (w,X,x) , where X = { x _kn } (-∞,∞) is an interpolatory matrix. We prove that for arbitrary X (-∞,∞) and ɛ > 0 , fixed, λ _n (w, X, x) ≥ c ɛ log n, x ∈ -a _n , a _n ]\H _n , n ≥ 1, where a _n is the MRS number and |H _n | ≤ 2 ɛ 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. October 28, 1996. Date revised: April 7, 1997. Date accepted: March 18, 1998.