On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

For n 1, let fxjngnj=1 be n distinct points and let Ln[ ] denote the corresponding Lagrange Interpolation operator. Let W : R ! [0;1). 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 ])W b kLp(R)= 0 for every continuous f : R ! Rwith suitably restricted growth, and some “weighting factor” ? We obtain a necessary and su¢ cient condition for such a p to exist. The result is the weighted analogue of our earlier work for interpolation arrays contained in a compact set. 1. The Result While there are very many results on mean convergence of Lagrange interpolation, the vast majority of these results deal with interpolation at zeros of orthogonal polynomials and their close cousins at least in terms of su¢ cient conditions for mean convergence see [3], [5], [6], [9]. In a recent paper [2], the author used distribution functions to treat general interpolation arrays contained in a compact set. Here we consider the non-compact case, and use decreasing rearrangements of functions, as well as a well known inequality of Hardy and Littlewood. Throughout, we consider an arrayX of interpolation pointsX = fxjng1 j n; n 1 where 1 < xnn < xn 1;n < < x2n < x1n <1: We denote by Ln[ ] the associated Lagrange interpolation operator, so that for f : R! R, we have Ln[f ](x) = n X