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An Introduction to Orthogonal Polynomials

Theodore S. Chihara
Publisher: 
Dover Publications
Publication Date: 
2011
Number of Pages: 
249
Format: 
Paperback
Price: 
19.95
ISBN: 
9780486479293
Category: 
Textbook
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
11/22/2011
]

This is a very specialized textbook giving an axiomatic treatment of Orthogonal Polynomial Sequences (defined below), slanted heavily toward the classical moment problem and toward recurrences defining the polynomials. This class covers most of the polynomials used in mathematical physics and differential equations, such as the Legendre polynomials and Hermite polynomials, although the approach here is not from the eigenfunction-expansion viewpoint as it would be in those subjects. The present work is an unaltered reprint of the 1978 edition from Gordon and Breach.

An Orthogonal Polynomial Sequence (OPS) is a sequence of polynomials Pn(x) such that Pn has degree n and any two polynomials are orthogonal. Here the inner product is defined in terms of a given linear functional L, so that L(Pn Pm) = 0 if and only if n ≠m. Typically the functional would be defined by multiplying its argument by a fixed weight function and integrating over a fixed interval, but whether the functional has this specific form is unimportant for most of the theory.

The book is written for independent study, on the very practical ground that there are never any courses in this subject. The first two-thirds of the book deals with the general theory, and covers moments, recurrences, behavior of zeroes, quadrature formulas, and continued fractions. For the most part the exposition remains general, with discussions of particular OPSs covered as exercises. The last third of the book is a catalog of all the different OPSs of interest that summarizes their most important properties, but does not give proofs. The writing is very clear, the prerequisites are a fairly small subset of undergraduate mathematics, and there is a thorough subject index and symbol glossary.

Another good book on the subject, but very different, is Dunham Jackson’s Carus Monograph Fourier Series and Orthogonal Polynomials. Jackson’s book focuses on the differential-equation and convergence aspects of the subject and is much more concrete than Chihara’s. The classical work in this field is Szegö’s 1939 AMS Colloquium Publication Orthogonal Polynomials. This is more encyclopedic than Chihara or Jackson, but is closer to Jackson in approach and in particular does not deal at all with the moment problem.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

Preface
I. Elementary Theory of Orthogonal Polynomials
II. The Representation Theorem and Distribution
III. Continued Fractions and Chain Sequences
IV. The Recurrence Formula and Properties of Orthogonal Polynomials
V. Special Functions
VI. Some Specific Systems of Orthogonal Polynomials
Notes
Appendix Table of Recurrence Formulas
List of Frequently Used Symbols
Bibliography
Index

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