Marcinkiewicz-Zygmund Inequalities: Methods and Results

Abstract

The Gauss quadrature formula for a weight W 2 on the realline has the form tAjnP(Xjn) = ! PW 2 j=l for polynomials P of degree :5 2n-1. In studying eonvergenee of Lagrange interpolation in L p norms, p =1= 2, one needs forward and eonverse quadrature sum estimates such as t AjnW-2 (Xjn)IPWI P (xjn) :5~ C ! IPWI P j=l with C independent of n and P. These are often ealled Marcinkiewicz-Zygmund inequalities after their founders. We survey methods to prove these and the results that ean he achieved using them. Our foeus is on weights on the whole real line, hut we also refer to results for (-1,1) and the plane. In partieular, we present four methods to prove forward estimates and two to prove eonverse ones.