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# Fourier Series and Orthogonal Polynomials

DUNHAM JACKSON
Volume: 6
Copyright Date: 1941
Edition: 1
Pages: 249
https://www.jstor.org/stable/10.4169/j.ctt5hh8cf

## Table of Contents

1. Front Matter
(pp. i-iv)
2. PREFACE
(pp. v-viii)
Dunham Jackson
3. Table of Contents
(pp. ix-xiv)
4. CHAPTER I FOURIER SERIES
(pp. 1-44)

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

(1)$f(x) = {A_0} + {a_1}\cos x + {a_2}\cos 2x + ... + {b_1}\sin x + {b_2}\sin 2x + ...$

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

Since each term is a periodic function with period 2π, the sum of the series necessarily has the same period. (A function$\phi (x)$is said to have a constantaas a period if$\phi (x+a)$is identically equal to$\phi (x)$, even thoughamay not be the smallest value for which a relation of this sort is satisfied. If...

5. CHAPTER II LEGENDRE POLYNOMIALS
(pp. 45-68)

A type of series resembling Fourier series in many respects is one in which the sines and cosines are replaced by certain polynomials, called Legendre polynomials. These can be defined in a variety of ways, and their principal properties are so interrelated that the arrangement of them in logical sequence admits a multiplicity of permutations. The order of exposition followed here for a rapid derivation of the principal facts is chosen rather for its convenience from the point of view of the results to be obtained than for any obvious motivation of its steps in advance. One of the possible...

6. CHAPTER III BESSEL FUNCTIONS
(pp. 69-90)

Here, as in the preceding chapter, certain facts will be presented in logical sequence without much emphasis on motivation at the outset. An obvious bond uniting this chapter with the two preceding is the property of orthogonality of the functions concerned. An organization of ideas under which Fourier, Legendre, and Bessel series appear not merely with common characteristics, but with a common origin, will be set forth in the next two chapters, on boundary value problems. A distinguishing feature of the series to be dealt with at present, from the point of view of elementary approach, is that they involve...

7. CHAPTER IV BOUNDARY VALUE PROBLEMS
(pp. 91-114)

The series of Fourier, Legendre, and Bessel, together with others, have a common field of application in connection with the solution of what are commonly called the “partial differential equations of mathematical physics.” One of the most important of these isLaplace’s equation, having for three independent variables the form

(1)$\frac{{{\partial ^2}u}} {{\partial {x^2}}} + \frac{{{\partial ^2}u}} {{\partial {y^2}}} + \frac{{{\partial ^2}u}} {{\partial {z^2}}} = 0$

No less important is the corresponding equation in two independent variables,

(2)$\frac{{{\partial ^2}u}} {{\partial {x^2}}} + \frac{{{\partial ^2}u}} {{\partial {y^2}}} = 0$

to which (1) reduces either if a plane problem is under consideration instead of one in space, or if the functions Relating to a space problem are so specialized as to be independent of the...

8. CHAPTER V DOUBLE SERIES; LAPLACE SERIES
(pp. 115-141)

More general boundary value problems than those of the last chapter call for the expansion of functions of two independent variables in series of special functions which are orthogonal with respect to integration over a two-dimensional domain. One type of such series, called Laplace series, is of highly distinctive character, and will be studied in some detail below. An example will be presented first which leads to a form of series with less striking features of novelty.

Let a function$f(x,y)$be defined for$0\underline{\underline < } x\underline{\underline < } \pi$,$0\underline{\underline < } y\underline{\underline < } \pi$, and let a boundary value problem be formulated for a cube of corresponding dimensions...

9. CHAPTER VI THE PEARSON FREQUENCY FUNCTIONS
(pp. 142-148)

The subject matter of this chapter has at first sight no obvious connection with what has gone before, but is introduced here to serve as a background for subsequent developments (see §10 in Chapter VII).

A class of functions proposed by Karl Pearson for the representation of statistical frequencies, and extensively used for that purpose, is defined by the differential equation

(1)$\frac{1} {y}\frac{{dy}} {{dx}} = \frac{{D + {E_x}}} {{A + Bx + C{x^2}}}$

in whichA, B, C, D, Eare constants. The “normal” frequency function${e^{ - {x^2}/2}}$satisfies this equation withE= -1,A= 1,B=C=D= 0. The pearson functions may be regarded...

10. CHAPTER VII ORTHOGONAL POLYNOMIALS
(pp. 149-165)

It is seen by reference to §3 of Chapter V that

$\int_{ - 1}^1 {(1 - {x^2})} P_k^1(x)P_k^1(x)dx = 0$

if${P_m}(x)$,${P_k}(x)$are Legendre polynomials of different degrees. This relation can be described by saying that the functions${(1 - {x^2})^{1/2}}P_m^1(x),{(1 - {x^2})^{1/2}}P_k^1(x)$are orthogonal to each other over the interval (—1, 1). It can be alternatively expressed, in a more significant manner for the following discussion, by saying that the polynomials$P_m^1(x)$,$P_k^1(x)$themselves are orthogonalwith respect to the weight function$1-{x^2}$. More generally, the polynomials$P_m^{(n)}(x)$,$P_k^{(n)}(x)$are orthogonal with respect to${(1-{x^2})^n}$as weight function, ifnis any positive integer. The concept of systems of...

11. CHAPTER VIII JACOBI POLYNOMIALS
(pp. 166-175)

The domain of orthogonality of the Jacobi polynomials is a finite interval, which may without essential loss of generality be taken as (—1, 1), since an arbitrary finite interval can be reduced to this by a linear change of variable. The weight function is

$p(x) = {(1 - x)^\alpha }{(1 + x)^\beta }$,

in which the exponents are arbitrary real numbers satisfying the condition that$\alpha > - 1,\beta > - 1$, this restriction being imposed so that$p(x)$shall be integrable from —1 to 1. The orthogonal polynomials will be defined outright by means of a derivative formula generalizing the Rodrigues formula for the Legendre polynomials. The property of orthogonality will be...

12. CHAPTER IX HERMITE POLYNOMIALS
(pp. 176-183)

The Hermite polynomials are orthogonal over the interval (—∞, ∞). The weight function* is${e^{ - {x^2}/2}}$.

Let

$\phi (x) = {e^{ - {x^2}/2}}$

By straightforward differentiation,

$\phi '(x) = - x{e^{ - {x^2}/2}}$,$\phi ''(x) = ({x^2} - 1){e^{ - {x^2}/2}}$,$\phi '''(x) = ( - {x^3} + 3x){e^{ - {x^2}/2}},...$

It is clear by inspection, and will presently be apparent on the basis of a more formal induction, that the derivative of any order is the product of${e^{ - {x^2}/2}}$by a polynomial inx.

Let

${H_n}(x) = {( - 1)^n}{e^{{x^2}/2}}\frac{{dn}} {{d{x^n}}}\phi (x)$

Then${\phi ^{(n)}}(x) = {( - 1)^n}{e^{ - {x^2}/2}}{H_n}(x)$, and by differentiation of this identity

${\phi ^{(n + 1)}}(x) = {( - 1)^n}[ - x{H_n}(x) + H_n^1(x)]{e^{ - {x^2}/2}}$,

while on the other hand${\phi ^{(n + 1)}}(x) = {( - 1)^{n + 1}}{e^{ - {x^2}/2}}{H_{n + 1}}(x)$, so that

(1)${H_{n + 1}}(x) = x{H_n}(x) - H_n^1(x)$.

Since${H_0}(x) = 1$it follows by induction that${H_n}(x)$is a polynomial of thenth degree with the coefficient of${x_n}$equal...

13. CHAPTER X LAGUERRE POLYNOMIALS
(pp. 184-190)

The interval of orthogonality of the Laguerre polynomials is (0, ∞). The weight function is

$p(x) = {x^\alpha }{e^{ - x}}$,

in which, for the sake of integrability, it is assumed that$\alpha > - 1$. This is a generalization of the simple exponential weight function${e^{ - x}}$, with which the name is sometimes more restrictively associated.

Let

${\phi _n}(x) = {x^{\alpha + n}}{e^{ - x}}$.

By application of Leibniz’s formula to thenth derivative of this product it is apparent that$\phi _n^{(n)}(x)$is the product of${x^\alpha }{e^{ - x}}$by a polynomial of thenth degree having${( - 1)^n}$as coefficient of${x^n}$. Let this polynomial multiplied by${( - 1)^n}$be denoted by$L_n^{(\alpha )}(x)$or more simply${L_n}(x)$:

(1)...

14. CHAPTER XI CONVERGENCE
(pp. 191-208)

The question of the convergence of series of orthogonal polynomials under the most general hypotheses, as might be anticipated, is one of great complexity. A considerable part of the theory of convergence developed for Fourier series in the first chapter, or for Legendre series in the second, can however be extended with little additional difficulty to orthogonal polynomials corresponding to various weight functions on a finite intervalif the polynomials of the orthonormal set are bounded as n becomes infinite,i.e., if there is a constantH, independent ofn, such that$\left| {{p_n}(x)} \right|\underline \leqslant H$for all values ofn, either at...

15. EXERCISES
(pp. 209-228)
16. BIBLIOGRAPHY OF SUGGESTIONS FOR SUPPLEMENTARY READING
(pp. 229-230)
17. INDEX OF NAMES
(pp. 231-234)