The weighted Lp-norms of orthonormal polynomials for Erdös weights

Let W := e−Q, where Q: → is even, and “smooth,” and of faster than polynomial growth at infinity. For example, we consider Q(x) = expκl(xα), α > 1, where expκ = exp(exp(...exp(...))) denotes the kth iterated exponential. Weights of the form W2 for such W are often called Erdös weights. We compute the growth of the Lp-norms (0 < p < ∞) of the weighted orthonormal polynomials pn(W2,x)W(x) for a large class of Erdös weights, based on recent work of the author with Levin and Mthembu on the L∞-norm of pn(W2,x)W(x). As an auxiliary result, we obtain bounds on the fundamental polynomials of Lagrange interpolation at the zeros of pn(W2,x), and as a corollary, we deduce finer spacing for the zeros of pn(W2•). The growth of the Lp-norms of orthonormal polynomials is a key factor in investigating convergence of orthogonal expansions and Lagrange interpolation in Lp-norms.