General Orthogonal Polynomials

An encyclopedic presentation of general orthogonal polynomials, placing emphasis on asymptotic behaviour and zero distribution.

Herbert Stahl (Author), Vilmos Totik (Author)

9780521415347, Cambridge University Press

Hardback, published 24 April 1992

268 pages
24.3 x 16.3 x 2 cm, 0.512 kg

Review of the hardback: 'This very clearly written book can be warmly recommended.' Acta Sci. Math.

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.

Introduction
1. Upper and lower bounds
2. Zero distribution of orthogonal polynomials
3. Regular n-th root asymptotic behaviour of orthogonal polynomials
4. Regularity criteria
5. Localization
6. Applications
Appendix
Notes and bibliographical references
Bibliography
List of symbols
Index.

Subject Areas: Calculus & mathematical analysis [PBK]