Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r > 0, b ∈ (− ∞, 2]. We establish a full quadrature sum estimate ∑j=1nλjn|PW|p(xjn)W−b↬C∫−∞∞|PW|p(t)W2−b(t)dt1 ⩽ p < ∞, for every polynomial P of degree at most n + rn13, where W2 is a Freud weight such as exp(−¦x¦α), α > 1, λjn are the Christoffel numbers, xjn are the zeros of the orthonormal polynomials for the weight W2, and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree ⩽ m = m(n) if m(n) = n + ξnn13, where ξn → ∞ as n → ∞. Previous estimates could sum only over those xjn with ¦xjn¦ ⩽ σx1n, some fixed 0 < σ < 1.