Full quadrature sums for pth powers of polynomials with Freud weights

Abstract

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=1 n λ jn|PW| p(x jn)W −b↬C ∫ −∞ ∞ |PW| p(t)W 2−b(t) dt 1 ⩽ p < ∞, for every polynomial P of degree at most n + rn 1 3 , where W 2 is a Freud weight such as exp(−¦x¦ α) , α > 1, λ jn are the Christoffel numbers, x jn are the zeros of the orthonormal polynomials for the weight W 2, 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 + ξ nn 1 3 , where ξ n → ∞ as n → ∞. Previous estimates could sum only over those x jn with ¦x jn¦ ⩽ σx 1n , some fixed 0 < σ < 1.