On mean convergence of Lagrange interpolation for general arrays

For n greater than or equal to 1, let {X-jn}(j=1)(n) be n distinct points in a compact set K is an element of R and let L-n[.] denote the corresponding Lagrange interpolation operator. Let r be a suitably restricted function on K. What conditions on the array {x(m)}(1 less than or equal to j less than or equal to n, n greater than or equal to 1) ensure the existence of p > 0 such that lim(n-->f) parallel to(f- L-n[f]) v parallel to L-F(K) = 0 for very continuous f: K--> R? We show that it is necessary and sufficient that there crisis r>0 with sup(n greater than or equal to 1) parallel to pi(n)nu parallel to L-f(K) Sigma(j=1)(n) (1//pi(n)'/(x(jn))) < infinity. Here for n greater than or equal to 1, pi(n) is a polynomial of degree n having {x(jn)}(j=1)(n) as zeros. The necessity of this condition is due to Ying Guang Shi. (C) 2000 Academic Press.