Necessary and Sufficient Conditions for Mean Convergence of Lagrange Interpolation for Erdős Weights

Abstract

Abstract We investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials p n ( W 2 , x ) for Erdös weights W 2 = e -2 Q . The archetypal example is W k,α = exp(— Q k,α ), where α > 1, k ≥ 1, and is the k -th iterated exponential. Following is our main result: Let 1 < p < ∞, Δ ∊ ℝ, k > 0. Let L n [ f ] denote the Lagrange interpolation polynomial to ƒ at the zeros of p n ( W 2 , x ) = p n ( e -2 Q , x ). Then for to hold for every continuous function ƒ : ℝ —> ℝ satisfying it is necessary and sufficient that