On Mean Convergence of Lagrange Interpolation for General Arrays

For n 1, let fxjngnj=1 be n distinct points in a compact set K R and let Ln[ ] denote the corresponding Lagrange Interpolation operator. Let v be a suitably restricted function on K. What conditions on the array fxjng1 j n; n 1 ensure the existence of p > 0 such that lim n!1 k (f Ln[f ]) v kLp(K)= 0 for every continuous f :: K ! R ? We show that it is necessary and su cient that there exists r > 0 with sup n 1 k nv kLr(K) n X j=1 1 j 0 nj (xjn) <1: Here for n 1; n is a polynomial of degree n having fxjngnj=1 as zeros. The necessity of this condition is due to Ying Guang Shi. 1 The Result There is a vast literature on mean convergence of Lagrange interpolation, based primarily at zeros of orthogonal polynomials and their close cousins. See [3 { 10] for recent references. Most of the work dealing with mean convergence of Lagrange interpolation for general arrays involves necessary conditions [6], [9], since su cient conditions are hard to come by. Some su cient conditions for convergence of general arrays in Lp; p > 1, have been given in [3]. In a recent paper, the author showed that distribution functions and Loomis' Lemma may be used to investigate mean convergence of Lagrange interpolation in Lp; p < 1 [2]. Indeed those techniques show that investigating convergence of Lagrange interpolation in Lp is inherently easier for p < 1 than for p 1. Here we show that similar ideas may be used to solve the problem of whether there is convergence in weighted Lp spaces for at least one p > 0.