Necessary and sufficient conditions for mean convergence of Lagrange interpolation for Erdos weights

We investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials p(n)(W-2,x) for Erdos weights W-2 = e(-2Q). The archetypal example is W-k,W-alpha = exp(-Q(k,alpha)), where Q(k,alpha)(x) := exp(k)(\x\(alpha)), alpha > 1, k greater than or equal to 1, and exp(k) = exp(exp(exp(...))) is tho k-th iterated exponential. Following is our main result: Let 1 < p < infinity, Delta epsilon R, kappa > 0. Let L(n)[f] denote the Lagrange interpolation polynomial to f at the zeros of p(n)(W-2,x) = p(n)(e(-2Q),x). Then for [GRAPHICS] to hold for every continuous function f: R --> R satisfying [GRAPHICS] it is necessary and sufficient that Delta > max{0, 2/3(1/4 - 1/p)}.