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  • Print publication year: 1992
  • Online publication date: August 2010

Preface

Summary

The theory of orthogonal polynomials can be divided into two loosely related parts. One of them is the formal, algebraic aspect of the theory, which has close connections with special functions, combinatorics, and algebra, and it is mainly devoted to concrete orthogonal systems or hierarchies of systems such as the Jacobi, Hahn, Askey-Wilson, … polynomials.

The investigation of more general orthogonal polynomials with methods of mathematical analysis belongs to the other part of the theory. Here the central questions are the asymptotic behavior of the polynomials and their zeros, recovering the measure of orthogonality, and so forth. This part has applications to approximation processes such as polynomial and rational interpolation, Padé approximation, and best rational approximation, to Fourier expansions, quadrature processes, eigenvalue problems, and so forth.

Textbooks on orthogonal polynomials usually cover material from both parts of the theory but give different emphasis in accordance with individual preference. Only the classical book [Sz3] by Gábor Szegő aims at a treatment of the subject in an encyclopedic manner. The present book is exclusively devoted to the second part of the theory. The main emphasis is on the investigation of the asymptotic behavior of general orthogonal polynomials, but related questions as, for instance, the distribution of zeros are also taken into consideration. A whole chapter is devoted to applications of the results in other areas.

Until now most of the asymptotic theory of orthogonal polynomials has concentrated on orthogonal systems for which the measure of orthogonality is supported on the real line or on the unit circle.