Strong Asymptotics For Extremal Polynomials Associated With Weights On R

The aim of this research monograph is to establish strong, or SzegA type asymptotics for extremal polynomials associated with weights W(x): = exp (-Q(x)) on . While the Q(x) treated are fairly general - even and of smooth polynomial growth at infinity - a typical example is Q(x): =, > 0. The results are consequences of a strengthened form of the following assertion: Given 0

, f Lp ( ) and a certain sequence of positive numbers associated with Q(x), there exist polynomials Pn of degree at most n, n = 1,2,3..., such that if and only if f(x) = 0 for a.e. ">" 1. Auxiliary results include inequalities for weighted polynomials, and zeros of extremal polynomials. The monograph is fairly self-contained, with proofs involving elementary complex analysis, and the theory of orthogonal and extremal polynomials. It should be of interest to research workers in approximation theory and orthogonal polynomials.