Page:EB1911 - Volume 11.djvu/321

n＝∞. The convergence is said to be “uniform” in an interval if, after specification of ε, the same number n suffices at all points of the interval to make |ƒ(x) − ƒ_m(x)| < ε for all values of m which exceed n. The numbers n corresponding to any ε, however small, are all finite, but, when ε is less than some fixed finite number, they may have an infinite superior limit (§ 7); when this is the case there must be at least one point, a, of the interval which has the property that, whatever number N we take, ε can be taken so small that, at some point in the neighbourhood of a, n must be taken > N to make |ƒ(x) − ƒ_m(x)| < ε when m > n; then the series does not converge uniformly in the neighbourhood of a. The distinction may be otherwise expressed thus: Choose a first and ε afterwards, then the number n is finite; choose ε first and allow a to vary, then the number n becomes a function of a, which may tend to become infinite, or may remain below a fixed number; if such a fixed number exists, however small ε may be, the convergence is uniform.

For example, the series sin x−1/2 sin 2x＋1/3 sin 3x−. . . is convergent for all real values of x, and, when π > x > −π its sum is 1/2x; but, when x is but a little less than π, the number of terms which must be taken in order to bring the sum at all near to the value of 1/2x is very large, and this number tends to increase indefinitely as x approaches π. This series does not converge uniformly in the neighbourhood of x＝π. Another example is afforded by the series $\sum _{n=0}^{\infty }{\frac {nx}{n^{2}x^{2}+1}}-{\frac {(n+1)x}{(n+1)^{2}x^{2}+1}}$, of which the remainder after n terms is nx/(n²x² + 1). If we put x＝1/n, for any value of n, however great, the remainder is 1/2; and the number of terms required to be taken to make the remainder tend to zero depends upon the value of x when x is near to zero—it must, in fact, be large compared with 1/x. The series does not converge uniformly in the neighbourhood of x＝0.

As regards series whose terms represent continuous functions we have the following theorems:

(1) If the series converges uniformly in an interval it represents a function which is continuous throughout the interval.

(2) If the series represents a function which is discontinuous in an interval it cannot converge uniformly in the interval.

(3) A series which does not converge uniformly in an interval may nevertheless represent a function which is continuous throughout the interval.

(4) A power series converges uniformly in any interval contained within its domain of convergence, the end-points being excluded.

(5) If $\sum _{r=0}^{\infty }f_{r}(x)=f(x)$ converges uniformly in the interval between a and b

$\int _{a}^{b}f(x)dx=\sum _{r=0}^{\infty }\int _{a}^{b}f_{r}(x)dx,$

or a series which converges uniformly may be integrated term by term.

(6) If $\sum _{r=0}^{\infty }f'_{r}(x)$ converges uniformly in an interval, then $\sum _{r=0}^{\infty }f_{r}(x)$ converges in the interval, and represents a continuous differentiable function, φ(x); in fact we have

$\phi ^{'}(x)=\sum _{r=0}^{\infty }f'_{r}(x)$,

or a series can be differentiated term by term if the series of derived functions converges uniformly.

A series whose terms represent functions which are not continuous throughout an interval may converge uniformly in the interval. If $\sum _{r=0}^{\infty }f_{r}(x),=f(x)$ is such a series, and if all the functions ƒ_r(x) have limits at a, then ƒ(x) has a limit at a, which is $\sum _{r=0}^{\infty }$ Lt
x＝a $f_{r}(x)$. A similar theorem holds for limits on the left or on the right.

23. Fourier’s Series.—An extensive class of functions admit of being represented by series of the form

$a_{0}+\sum _{n=1}^{\infty }{\big (}a_{n}\cos {\frac {n\pi x}{c}}+b_{n}\sin {\frac {n\pi x}{c}}{\big )},$

(i.)

and the rule for determining the coefficients a_n, b_n of such a series, in order that it may represent a given function ƒ(x) in the interval between −c and c, was given by Fourier, viz. we have

$a_{0}={\frac {1}{2c}}\int _{-c}^{c}f(x)dx,a_{n}={\frac {1}{c}}\int _{-c}^{c}f(x)\cos {\frac {n\pi x}{c}}dx,b_{n}={\frac {1}{c}}\int _{-c}^{c}f(x)\sin {\frac {n\pi x}{c}}dx.$

The interval between −c and c may be called the “periodic interval,” and we may replace it by any other interval, e.g. that between 0 and 1, without any restriction of generality. When this is done the sum of the series takes the form

Lt
n＝∞ $\int _{0}^{1}\sum _{r=-n}^{r=n}f(z)\cos\{2r\pi (z-x)\}dz,$

and this is

Lt n＝∞ $\int _{0}^{1}f(z){\frac {\sin\{(2n+1)(z-x)\pi \}}{\sin\{(z-x)\pi }}dz.$

(ii.)

Fourier’s theorem is that, if the periodic interval can be divided into a finite number of partial intervals within each of which the function is ordinary (§ 14), the series represents the function within each of those partial intervals. In Fourier’s time a function of this character was regarded as completely arbitrary.

By a discussion of the integral (ii.) based on the Second Theorem of the Mean (§ 15) it can be shown that, if ƒ(x) has restricted oscillation in the interval (§ 11), the sum of the series is equal to 1/2{ƒ(x + 0) + ƒ(x − 0)} at any point x within the interval, and that it is equal to 1/2 {ƒ(+0) + ƒ(1 − 0} at each end of the interval. (See the article Fourier’s Series.) It therefore represents the function at any point of the periodic interval at which the function is continuous (except possibly the end-points), and has a definite value at each point of discontinuity. The condition of restricted oscillation includes all the functions contemplated in the statement of the theorem and some others. Further, it can be shown that, in any partial interval throughout which ƒ(x) is continuous, the series converges uniformly, and that no series of the form (i), with coefficients other than those determined by Fourier’s rule, can represent the function at all points, except points of discontinuity, in the same periodic interval. The result can be extended to a function ƒ(x) which tends to become infinite at a finite number of points a of the interval, provided (1) ƒ(x) tends to become determinately infinite at each of the points a, (2) the improper definite integral of ƒ(x) through the interval is convergent, (3) ƒ(x) has not an infinite number of discontinuities or of maxima or minima in the interval.

24. Representation of Continuous Functions by Series.—If the series for ƒ(x) formed by Fourier’s rule converges at the point a of the periodic interval, and if ƒ(x) is continuous at a, the sum of the series is ƒ(a); but it has been proved by P. du Bois Reymond that the function may be continuous at a, and yet the series formed by Fourier’s rule may be divergent at a. Thus some continuous functions do not admit of representation by Fourier’s series. All continuous functions, however, admit of being represented with arbitrarily close approximation in either of two forms, which may be described as “terminated Fourier’s series” and “terminated power series,” according to the two following theorems:

(1) If ƒ(x) is continuous throughout the interval between 0 and 2π, and if any positive number ε however small is specified, it is possible to find an integer n, so that the difference between the value of ƒ(x) and the sum of the first n terms of the series for ƒ(x), formed by Fourier’s rule with periodic interval from 0 to 2π, shall be less than ε at all points of the interval. This result can be extended to a function which is continuous in any given interval.

(2) If ƒ(x) is continuous throughout an interval, and any positive number ε however small is specified, it is possible to find an integer n and a polynomial in x of the nth degree, so that the difference between the value of ƒ(x) and the value of the polynomial shall be less than ε at all points of the interval.

Again it can be proved that, if ƒ(x) is continuous throughout a given interval, polynomials in x of finite degrees can be found, so as to form an infinite series of polynomials whose sum is equal to ƒ(x) at all points of the interval. Methods of representation of continuous functions by infinite series of rational fractional functions have also been devised.

Particular interest attaches to continuous functions which are not differentiable. Weierstrass gave as an example the function represented by the series $\sum _{n=0}^{\infty }a^{n}\cos(b^{n}x\pi )$, where a is positive and less than unity, and b is an odd integer exceeding (1 + 3/2π)/a. It can be shown that this series is uniformly convergent in every interval,