On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

AbstractFor 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 limn→∞∥(f−Ln[f])Wφb∥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