On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

For n ≥ 1, let {xjn}j=1n be n distinct points and let Ln[.] denote the corresponding Lagrange Interpolation operator. Let W : R → [0, ∞). What conditions on the array {xjn}1 ≤ j ≤ n, n ≥ 1 ensure the existence of p > 0 such that limn→∞ || (f - Ln[f])W φb ||Lp(R) = 0 for every continuous f : R → R with suitably restricted growth, and some "weighting factor" φb? We obtain a necessary and sufficient 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.