Converse Quadrature Sum Inequalities for Freud Weights II

LetW := exp( Q), where Q is of smooth polynomial growth at1, for example Q(x) = jxj ; > 1. We call W 2 a Freud weight. Let fxjngnj=1 and f jngnj=1 denote respectively the zeros of the nth orthonormal polynomial pn for W 2 and the Christo¤el numbers of order n. We establish converse quadrature sum inequalities associated with W , such as k (PW ) (x) (1 + jxj) kLp(R) C n X j=1 jnW 2 (xjn) jPW j (xjn) (1 + jxjnj) with C independent of n and polynomials P of degree < n, and suitable restrictions on r;R. We concentrate on the case p 4, as the case p < 4 was handled earlier. Moreover, we are able to treat a general class of Freud weights, whereas our earlier treatment dealt essentially with exp jxj ; = 2; 4; 6; ::: . Some applications to Lagrange interpolation are presented. 1 Introduction and Results Let W := e , where Q : R ! R is even, convex, and is of polynomial growth at 1. We call W 2 a Freud weight. Corresponding to W , we can de…ne orthonormal polynomials pn(x) = nx n + :::; n = n(W ) > 0,