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 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.
From the reviews:
MATHEMATICAL REVIEWS
"This book is a must for approximators, in particular those interested in weighted polynomial approximation or orthogonal polynomials. It cannot serve as a textbook but will probably be indispensable for research in this field, since all the important tools, results, and properties are there, with detailed proofs and appropriate references."
"This book is the result of almost 2 decades of collaboration of the two authors. The style is a succession of theorems and proofs, clearly written, introducing the necessary definitions and concepts where needed. A must for researchers in orthogonal polynomials or in approximation theory, or in any other field where orthogonal polynomials play an essential role." (Adhemar Bultheel, Belgian Mathematical Society, Vol. 10 (2), 2003)
"It is a magnum opus that brings together the substantial research over 18 years on this topic by Levin and Lubinsky, two leaders in the field. With its attention to new results ... the work is presented more in the form of a very substantial research paper rather than a book. It will undoubtedly be of considerable use to researchers in this area ... . There is no doubt that this book is a substantial contribution to the literature on orthogonal polynomials ... ." (Walter R. Bloom, The Australian Mathematical Society Gazette, Vol. 30 (4), 2003)
"This monograph provides a systematic treatment of the theory and applications of orthogonal polynomials with exponential weights. ... In conclusion, this monograph will be a significant addition to the existing monographs on Orthogonal Polynomials and related areas. It will be useful for researchers working in the fields of mathematical, physical, statistical and engineering sciences." (R. K. Saxena, Journal of the Indian Academy of Mathematics, Vol. 24 (2), 2002)
"The authors of this monograph have greatly contributed to the present state of the art of this subject, and in this book present a comprehensive account of it. Parallel to this, they update the results either in strength or in generality, and so most of them appear for the first time in the form given here." (G. López Lagomasimo, Bulletin of the London Mathematical Society, Vol. 35, 2003)
"It ... looks like a set of research papers, with very recent results and detailed proofs, using the latest available tools and techniques ... . This book is a must for approximators, in particularly those interested in weighted polynomial approximation or orthogonal polynomials. It ... will probably be indispensable for research in this field, since all the important tools, results, properties are there, with detailed proofs and to the point references." (Walther Van Assche, Zentralblatt MATH, Vol. 997 (22), 2002)
