Fourier Series and Orthogonal Polynomials

Dunham Jackson

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486438085
• eBook ISBN: 9780486154503
• Topic: Mathématiques
• Subtopic: Analyse fonctionnelle

This text for undergraduate and graduate students illustrates the fundamental simplicity of the properties of orthogonal functions and their developments in related series. Starting with a definition and explanation of the elements of Fourier series, the text follows with examinations of Legendre polynomials and Bessel functions. Boundary value problems consider Fourier series in conjunction with Laplace's equation in an infinite strip and in a rectangle, with a vibrating string, in three dimensions, in a sphere, and in other circumstances. An overview of Pearson frequency functions is followed by chapters on orthogonal, Jacobi, Hermite, and Laguerre polynomials, and the text concludes with a chapter on convergence. 1941 edition.

CHAPTER I

FOURIER SERIES

1. Definition of Fourier series. A given function f(x) can be represented, under hypotheses of considerable generality, by an infinite series in the form

(1)

Such a series, when the coefficients are determined in the manner to be described below, is called a Fourier series.

Since each term is a periodic function with period 2π, the sum of the series necessarily has the same period. (A function φ(x) is said to have a constant a as a period if φ(x+a) is identically equal to φ(x), even though a may not be the smallest value for which a relation of this sort is satisfied. If a is a period, any integral multiple of a is also a period. In accordance with this definition, cos nx and sin nx have the period 2π, although they have also the smaller period 2π/n.)

On the other hand, a Fourier series is sometimes useful for the representation of a given function in a single interval of length 2π, when the property of periodicity is of no concern except as it results incidentally from evaluation of the series outside the interval in which the function was originally defined.

The period 2π may be replaced by one of arbitrary length, as will be pointed out later, with no additional difficulty beyond a slight loss of simplicity in the formulas.

2. Orthogonality of sines and cosines. The determination of the coefficients depends on the evaluation of certain definite integrals involving the sines and cosines which enter into the terms of the series. In the first place, if n is an integer not zero,

(2)

The latter relation holds also for n = 0; in the former, setting n = 0 replaces the right-hand member by 2π.

Throughout the rest of this section, p and q will be understood to represent non-negative integers. Since

it follows by application of the above with n = p − q and again with n = p + q that

(3)

when p≠q. If q = p≠0, the integral of cos (p+q)x over the period interval is still zero, while the other term gives

Similarly, the identities

give

(4)

(5)

the last relation holding whether p and q are the same or different.

The vanishing of the integrals in (3), (4) and (5) is expressed in words by saying that any two of the functions 1, cos x, cos 2x, ⋯ , sin x, sin 2x, ⋯ are orthogonal to each other over the interval (−π, π)....

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• PREFACE
• Table of Contents
• CHAPTER I - FOURIER SERIES
• CHAPTER II - LEGENDRE POLYNOMIALS
• CHAPTER III - BESSEL FUNCTIONS
• CHAPTER IV - BOUNDARY VALUE PROBLEMS
• CHAPTER V - DOUBLE SERIES; LAPLACE SERIES
• CHAPTER VI - THE PEARSON FREQUENCY FUNCTIONS
• CHAPTER VII - ORTHOGONAL POLYNOMIALS
• CHAPTER VIII - JACOBI POLYNOMIALS
• CHAPTER IX - HERMITE POLYNOMIALS
• CHAPTER X - LAGUERRE POLYNOMIALS
• CHAPTER XI - CONVERGENCE
• EXERCISES
• BIBLIOGRAPHY OF SUGGESTIONS FOR - SUPPLEMENTARY READING
• INDEX OF NAMES
• TOPICAL INDEX