On the Lebesgue Function of Weighted Lagrange Interpolation. I. (Freud-Type Weights)

Abstract. For a wide class of Freud-type weights of form w = exp(-Q) we investigate the behavior of the corresponding weighted Lebesgue function λn(w,X,x) , where X = { xkn} $\subset$ (-∞,∞) is an interpolatory matrix. We prove that for arbitraryX $\subset$ (-∞,∞) and ɛ > 0 , fixed, λn(w, X, x)≥ c ɛ log n,x ∈[-an, an]\Hn,n ≥ 1, where an is the MRS number and |Hn| ≤ 2ɛan . The result corresponds to the behavior of the ``ordinary'' Lebesgue function in [-1,1] . Other exponential weights are considered in our subsequent paper.