We establish pointwise as well as uniform estimates for Lebesgue functions associated with a large class of Erdo s weights on the real line. An Erdo s weight is of the form W :=exp(&Q), where Q : R ร R is even and is of faster than polynomial growth at infinity. The archetypal examples are
where Q k, : (x) :=exp k (|x| : ), :>1, k 1. Here exp k :=exp(exp(exp(...))) denotes the k th iterated exponential.
where Q A, B (x) :=exp(log(A+x 2 )) B , B>1 and A>A 0 . For a carefully chosen system of nodes / n :=[! 1 , ! 2 , ..., ! n ], n 1, our result imply in particular, that the Lebesgue constant
Moreover, we show that this choice of nodes is optimal with respect to the zeros of the orthonormal polynomials generated by W 2 . Indeed, let U n :=[x j, n : 1 j n], n 1, where the x k, n are the zeros of the orthogonal polynomials p n (W 2 , } ) generated by W 2 . Then in particular, we have uniformly for n N, &4 n (W k, :
Here, log j :=log(log(log(...))) denotes the j th iterated logarithm. We deduce sharp theorems of uniform convergence of weighted Lagrange interpolation together with rates of convergence. In particular, these results apply to W k, : and W A, B .