General orthogonal polynomials / Herbert Stahl, Vilmos Totik.

Format:

Book

Author/Creator:

Stahl, Herbert, author.

Totik, V., author.

Series:

Encyclopedia of mathematics and its applications ; v. 43.

Encyclopedia of mathematics and its applications ; volume 43

Language:

English

Subjects (All):

Orthogonal polynomials.

Physical Description:

1 online resource (xii, 250 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 1992.

Language Note:

English

Summary:

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.

Contents:

Cover; Title; Copyright; Contents; Preface; Acknowledgments; Symbols; 1 Upper and Lower Bounds; 1.1 Statement of the Main Results; 1.2 Some Potential-theoretic Preliminaries; 1.3 Proof of the Upper and Lower Bounds; 1.4 Proof of the Sharpness of the Upper and Lower Bounds; 1.5 Examples; 2 Zero Distribution of Orthogonal Polynomials; 2.1 Zeros of Orthogonal Polynomials; 2.2 Norm Asymptotics and Zero Distribution; 2.3 Asymptotic Behavior of Zeros when cμ > 0; 3 Regular nth-root Asymptotic Behavior of Orthonormal Polynomials; 3.1 Regular Asymptotic Behavior

3.2 Characterization of Regular Asymptotic Behavior3.3 Regular Behavior in the Case of Varying Weights; 3.4 Characterization of Regular Asymptotic Behavior in LP(μ); 3.5 Examples; 3.6 Regular Behavior and Monic Polynomials; 4 Regularity Criteria; 4.1 Existing Regularity Criteria and Their Generalizations; 4.2 New Criteria and Their Sharpness; 4.3 Proof of the Regularity Criteria; 4.4 Preliminaries for Proving the Sharpness of the Criteria; 4.5 Proof of the Sharpness of the Regularity Criteria; 4.6 Summary of Regularity Criteria and Their Relations; 5 Localization

5.1 Global versus Local Behavior5.2 Localization at a Single Point; 5.3 Localization Theorems; 6 Applications; 6.1 Rational Interpolants to Markov Functions; 6.2 Best Rational Approximants to Markov Functions; 6.3 Nondiagonal Pade Approximants to Markov Functions; 6.4 Weighted Polynomials in Lp(μ); 6.5 Regularity and Weighted Chebyshev Constants; 6.6 Regularity and Best L2(μ) Polynomial Approximation; 6.7 Determining Sets; Appendix; A.I Energy and Capacity; A.II Potentials, Fine Topology; A.III Principles; A.IV Equilibrium Measures; A.V Green Functions; A.VI Dirichlet's Problem

A.VII BalayageA.VIII Green Potential and Condenser Capacity; A.IX The Energy Problem in the Presence of an External Field; Notes and Bibliographical References; Bibliography; Index

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-88638-X

1-107-10287-1

1-107-10032-1

0-521-13504-4

1-107-09454-2

1-107-08830-5

0-511-75942-8

1-107-09133-0

OCLC:

776951332