Full quadrature sums for pth powers of polynomials with Freud weights (original) (raw)

1995, Journal of Computational and Applied Mathematics

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. f s y, 2j. IPWIP(Xj,)W-b(Xj,) <~ C IPWIP(t) W2-b(t)dt, j=l-~ 1 ~< p < ~, for every polynomial P of degree at most n + rn 1/3, where W 2 is a Freud weight such as exp(-lx['), ~ > 1, {2j,} are the Christoffel numbers, {xj,} 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 = re(n) if re(n) = n + ~nn 1/3, where ¢. ~ ~ as n-* ~. Previous estimates could sum only over those x~, with Ixj, I <~ axe,, some fixed 0 < a < 1.