General Orthogonal Polynomials

Abstract

In this treatise, the authors present the general theory of orthogonal polynomials on the complex plane and several of its applications. The assumptions on the measure of orthogonality are general, the only restriction is that it has compact support on the complex plane. In the development of the theory the main emphasis is on asymptotic behaviour and the distribution of zeros. In the following chapters, the author explores the exact upper and lower bounds are given for the orthonormal polynomials and for the location of their zeros; regular n-th root asymptotic behaviour; and applications of the theory, including exact rates for convergence of rational interpolants, best rational approximants and non-diagonal Pade approximants to Markov functions (Cauchy transforms of measures). The results are based on potential theoretic methods, so both the methods and the results can be extended to extremal polynomials in norms other than L2 norms. A sketch of the theory of logarithmic potentials is given in an appendix.

Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product

“…Korovkin [9]. Actually, (2.7) follows from a still weaker condition known as Ullman's criterion given in [26] (see also Chapter 4 in [25]) in terms of minimal carrier capacity or an equivalent criterion of Widom [28]. Orthonormal polynomials for which (2.7) holds are said to have regular exterior asymptotic behaviour on [−1, 1].…”

Universality Limits at the Hard Edge of the Spectrum for Measures with Compact Support

“…We say that is regular on [ 1; 1] in the sense of Ullmann and/ or Stahl and Totik [28], if lim n!1 1=n n = 2: Theorem 1.1 Let be a …nite positive Borel measure on ( 1; 1) that is regular. Assume that for some > 0, is absolutely continuous in J = [1 ; 1], and in J, its absolutely continuous component has the form w = hw ( ; ) , where ; > 1: Assume that h (1) > 0 and h is continuous at 1.…”

Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

“…Thus we may choose B = 1 in Theorem 1.2. Moreover [21], the zero counting measures for the Legendre polynomial on [0; 1] converge weakly to dx p x (1 x) on (0; 1) ; so a corresponding statement is true for f n g on [ 1; 0]. By Theorem 1.2, the f n g converge weakly to a measure on [ 1; 0] with no mass point at 0.…”