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An Introduction to Orthogonal Polynomials
This concise introduction covers general elementary theory related to orthogonal polynomials and assumes only a first undergraduate course in real analysis. Topics include the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula and properties of orthogonal polynomials, special functions, and some specific systems of orthogonal polynomials. 1978 edition.
- GenresMathematics
272 pages, Paperback
First published December 31, 1978
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Displaying 1 - 3 of 3 reviews
July 22, 2012
A very nice introductory text to orthogonal polynomials. Written in 1978, it is still a good book to get the basics of orthogonal polynomials.
The book consists of six chapters.
Chapter one starts with the basic definition of an orthogonal polynomial system as a sequence of monic polynomials, one of every degree, which are orthogonal with respect to some moment functional. Then basic properties are proved such as the existence of a OPS, the recurrence formula, simplicity of the real zeros. Also Gauss quadrature formula is proved.
Chapter two basically proves Favard's theorem, using Gauss quadrature formula and Helly's Selection Principle (equivalent to the axiom of choice) to construct, under some assumptions, a measure which coincides with the moment functional.
Chapter three develops some basic tools which will be used a lot in chapter four. Chihara shows that there is an important connection between continued fractions and orthogonal polynomials systems.
Chapter four contains some nice standard theorems using the tools developed in chapter three. He shows that the coefficients in the recurrence relation do have connections with distribution of the zeros of the polynomials and at the end of the chapter he proves Blumenthal's and Krein's theorems.
Chapter five and six sum up some of the more important specific systems of orthogonal polynomials.
It was a nice book to read and gave me a lot of insight about the basic theory of orthogonal polynomials. However the last two chapters are a bit dry. Also a lot has happen is the last forty years and the theory in this book is far from complete. For example, the Askey scheme can not be found in this book. There is also no discussion about the q-analogue of the polynomials. This is completely understandable since the Askey scheme was found almost 17 years after this book.
I also did miss the connection with orthogonal polynomials and hypergeometric series, in which most orthogonal polynomials can be described very nicely.
I would suggest that reading the first four chapters is a good thing to get in to the basics of orthogonal polynomials. The last two chapters are of less important and it might be better to follow this up with something like Special Functions from Andrew, Askey and Roy. At this moment there is a reprint of this book available on amazon for aproximately 15 euros. For so little money this is a must buy.
The book consists of six chapters.
Chapter one starts with the basic definition of an orthogonal polynomial system as a sequence of monic polynomials, one of every degree, which are orthogonal with respect to some moment functional. Then basic properties are proved such as the existence of a OPS, the recurrence formula, simplicity of the real zeros. Also Gauss quadrature formula is proved.
Chapter two basically proves Favard's theorem, using Gauss quadrature formula and Helly's Selection Principle (equivalent to the axiom of choice) to construct, under some assumptions, a measure which coincides with the moment functional.
Chapter three develops some basic tools which will be used a lot in chapter four. Chihara shows that there is an important connection between continued fractions and orthogonal polynomials systems.
Chapter four contains some nice standard theorems using the tools developed in chapter three. He shows that the coefficients in the recurrence relation do have connections with distribution of the zeros of the polynomials and at the end of the chapter he proves Blumenthal's and Krein's theorems.
Chapter five and six sum up some of the more important specific systems of orthogonal polynomials.
It was a nice book to read and gave me a lot of insight about the basic theory of orthogonal polynomials. However the last two chapters are a bit dry. Also a lot has happen is the last forty years and the theory in this book is far from complete. For example, the Askey scheme can not be found in this book. There is also no discussion about the q-analogue of the polynomials. This is completely understandable since the Askey scheme was found almost 17 years after this book.
I also did miss the connection with orthogonal polynomials and hypergeometric series, in which most orthogonal polynomials can be described very nicely.
I would suggest that reading the first four chapters is a good thing to get in to the basics of orthogonal polynomials. The last two chapters are of less important and it might be better to follow this up with something like Special Functions from Andrew, Askey and Roy. At this moment there is a reprint of this book available on amazon for aproximately 15 euros. For so little money this is a must buy.
May 1, 2017
Not good as a reference - or at least not as good as Abramowitz and Stegun.
Not good for self-study: few exercises and no answers / worked problems.
No discussion of applications or Lie Algebra representations.
Not good for self-study: few exercises and no answers / worked problems.
No discussion of applications or Lie Algebra representations.
Displaying 1 - 3 of 3 reviews