Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses, it is also appropriate for independent study. 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. Numerous examples and exercises, an extensive bibliography, and a table of recurrence formulas supplement the text.
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Yes, you can access An Introduction to Orthogonal Polynomials by Theodore S Chihara in PDF and/or ePUB format, as well as other popular books in Mathematics & Applied Mathematics. We have over one million books available in our catalogue for you to explore.
It is an elementary exercise in calculus to use the trigonometric identity
to obtain the integration formula
The fact that the integral in (1.2) vanishes is expressed by saying that cos mθ and cos nθ are orthogonal over the interval (0, π) for m ≠ n. We also say that {1, cos θ, cos 2θ, …, cos nθ, …} is an orthogonal sequence over (0, π).
We observe that the change of variable, x = cos θ, converts (1.2) into
where we have written
We have
and by elementary trigonometric identities,
Using the identity (1.1) with m = 1, it is an easy proof by induction to show that Tn(x) is a polynomial in x of degree n. These polynomials are called the Tchebichef polynomials of the first kind. Because of (1.3) we say that the Tn(x) are orthogonal polynomials with respect to (1 – x2)–1/2, – 1 < x < 1. More precisely,
is an orthogonal polynomial sequence with respect to the weight function (1 – x2)–1/2 on the interval (–1, 1).
Somewhat more generally, consider a function w which is non-negative and integrable on an interval (a, b). We also assume that w(x) > 0 on...
Table of contents
Cover
Title Page
Copyright Page
Contents
Preface
Chapter I. Elementary Theory of Orthogonal Polynomials
Chapter II. The Representation Theorem and Distribution Functions
Chapter III. Continued Fractions and Chain Sequences
Chapter IV. The Recurrence Formula and Properties of Orthogonal Polynomials
Chapter V. Special Functions
Chapter VI. Some Specific Systems of Orthogonal Polynomials
