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=∞ ∫ 10 Σr = nr = −n ƒ(z) cos {2rπ(z − x)}dz,
and this is
| Lt n=∞ ∫ 10 ƒ(z) | sin {(2n + 1) (z − x)π} | dz. |
| sin {(z − x)π} |
(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 ½{ƒ(x + 0) + ƒ(x − 0)} at any point x within the interval, and that it is equal to ½ {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.