We complete our investigations of mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn
(W
2, x) for Erdős weights W
2 = e
-2Q
. The archetypal example is Wk,α
= exp(—Qk,α
), where
α > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < 4 and α ∊ ℝ Let Ln
[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn
(W
2, x) = pn
(e
-2Q
, x). Then for
to hold for every continuous function ƒ:ℝ. —> ℝ satisfying
it is necessary and sufficient that α > 1/p. This is, essentially, an extension of the Erdös-Turan theorem on L
2 convergence. In an earlier paper, we analyzed convergence for all p > 1, showing the necessity and sufficiency of using the weighting factor 1 + Q for all p > 4. Our proofs of convergence are based on converse quadrature sum estimates, that are established using methods of H. König.