Orthogonal Polynomials for Exponential Weights

The analysis of orthogonal polynomials associated with general
weights was a major theme in classical analysis in the twentieth
century, and undoubtedly will continue to grow in importance in
the future.
In this monograph, the authors investigate orthogonal polynomials for
exponential weights defined on a finite or infinite interval. The
interval should contain 0, but need not be symmetric about 0;
likewise the weight need not be even. The authors establish bounds
and asymptotics for orthonormal and extremal polynomials, and their
associated Christoffel functions. They deduce bounds on zeros of
extremal and orthogonal polynomials, and also establish Markov-
Bernstein and Nikolskii inequalities.
The book will be of interest to researchers in approximation theory,
harmonic analysis, numerical analysis, potential theory, and all
those that apply orthogonal polynomials.
The authors have collaborated actively since 1982 on various topics, and have published many joint papers, as well as a Memoir of the American Mathematical Society. The latter deals with a special case of the weights treated in this book. In many ways, this book is the culmination of 18 years of joint work on orthogonal polynomials,
drawing inspiration from the works of many researchers in the very active field of orthogonal polynomials.

Contents

• Introduction and Results
• Weighted Potential Theory: The Basics
• Basic Estimates for Q, a_t
• Restricted Range Inequalities
• Estimates for Measure and Potential
• Smoothness of /rho_t
• Weighted Polynomial Approximation
• Asymptotics of Extremal Errors
• Christoffel Functions
• Markov-Bernstein and Nikolskii Inequalities
• Zeros of Orthogonal Polynomials
• Bounds on Orthogonal Polynomials
• Further Bounds and Applications
• Asymptotics of Extremal Polynomials
• Asymptotics of Orthonormal Polynomials