Quadrature Sums Involving pth Powers of Polynomials

R. Askey’s problem of estimating quadrature sums involving pth powers of polynomials $(0 < p < \infty )$ in terms of integrals, is solved. In its simplest form, the method can extend the large sieve of number theory to sums involving pth powers, rather than just squares, of trigonometric polynomials. Further, the method yields estimates whenever the abscissas in the quadrature have a suitable spacing and the weights have suitable bounds. In particular, it may be applied to generalized Jacobi weights and to Freud weights.