On Mean Convergence of Lagrange Interpolation for General Arrays Original Research Article

For n⩾1, let {xjn}nj=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 {xjn}1⩽j⩽n, n⩾1 ensure the existence of p>0 such that limn→∞ ‖(f−Ln[f]) v‖Lp(K)=0 for very continuous f: K→R? We show that it is necessary and sufficient that there exists r>0 with supn⩾1 ‖πnv‖Lr(K) ∑nj=1 (1/|π′n| (xjn))<∞. Here for n⩾1, πn is a polynomial of degree n having {xjn}nj=1 as zeros. The necessity of this condition is due to Ying Guang Shi.