On Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}ⁿ_j=1 be n distinct points in a compact set K and letL_n·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {x_jn}_1jn, n1 ensure the existence of p>0 such that lim_n→∞ (f−L_nf]) v_Lp(K)=0 for very continuous f: K→ ? We show that it is necessary and sufficient that there exists r>0 with sup_n1 π_nv_Lr(K) ∑ⁿ_j=1 (1/|π′_n| (x_jn))<∞. Here for n1, π_n is a polynomial of degree n having {x_jn}ⁿ_j=1 as zeros. The necessity of this condition is due to Ying Guang Shi.