Fourier Series and Trigonometrical Series
9.1 Definition of Fourier Series.[1]
Series of the type \[ \begin{align*} \frac{1}{2} a_{\:\!0} + (a_{1} + \cos x + b_{1} \sin x) &+ (a_{2} + \cos 2x + b_{2} \sin 2x) + \cdots \\ & = \frac{1}{2} a_{\:\!0} + \sum_{n=1}^{\infty} ( a_{n} \cos n x + b_{n} \sin n x), \end{align*} \] where \( a_{n}\), \( b_{n}\) are independent of \( x\), are of great importance in many investigations. They are called trigonometrical series.[2]
If there is a function \(f(t)\) such that \(\int_{\!-\pi}^{\:\pi} f(t) \, d t\) exists as a Riemann integral or as an improper integral which converges absolutely, and such that \[ \pi a_{n} = \int_{-\pi}^{\pi}\! f(t) \cos nt \, d t, \quad \pi b_{n} = \int_{-\pi}^{\pi}\! f(t) \sin nt \, d t, \] then the trigonometrical series is called a Fourier series.
Trigonometrical series first appeared in analysis in connexion with the investigations of Daniel Bernoulli on vibrating strings; d’Alembert had previously solved the equation of motion \( \ddot{y} = a^{2} \dfrac{d^{2} y}{d x^{2}}\) in the form \(y = \frac{1}{2} \left\{ f(x+at) + f(x-at) \right\}\), where \(y=f(x)\) is the initial shape of the string starting from rest; and Bernoulli shewed that a formal solution is \[ y = \sum_{n=1}^{\infty} b_{n} \sin \frac{n \pi x}{l} \cos \frac{n \pi a t}{l}, \] the fixed ends of the string being \((0,0)\) and \((l,0)\); and he asserted that this was the most general solution of the problem. This appeared to d’Alembert and Euler to be impossible, since such a series, having period \(2l\), could not possibly represent such a function as \(c x \:\! (l-x)\) when \(t = 0\).[3] A controversy arose between these mathematicians, of which an account is given in Hobson’s Functions of a Real Variable.
Fourier, in his Theorie de la Chaleur, investigated a number of trigonometrical series and shewed that, in a large number of particular cases, a Fourier series actually converged to the sum \(f(x)\). Poisson attempted a general proof of this theorem, Journal de l’École polytechnique, xii. (1823), pp. 404–509. Two proofs were given by Cauchy, Mém. de l’Acad. R. des Sci. vi. (1823, published 1826), pp. 603–612 (Oeuvres, (1), ii. pp. 12–19) and Exercices de Math. ii. (1827), pp. 341–376 (Oeuvres, (2), vii. pp. 393–430); these proofs, which are based on the theory of contour integration, are concerned with rather particular classes of functions and one is invalid. The second proof has been investigated by Harnack, Math. Ann. xxxii. (1888), pp. 175–202.
In 1829, Dirichlet gave the first rigorous proof that, for a general class of functions, the Fourier series, defined as above, does converge to the sum \(f(x)\).[4] A modification of this proof was given later by Bonnet.[5]
The result of Dirichlet is that if \(f(t)\) is defined and bounded in the range \((-\pi, \pi)\) and if \(f(t)\) has only a finite number of maxima and minima and a finite number of discontinuities in this range and, further, if \(f(t)\) is defined by the equation \[ f(t + 2 \pi) = f(t) \] outside the range \((-\pi, \pi)\), then, provided that \[ \pi a_{n} = \int_{-\pi}^{\pi}\! f(t) \cos nt \, d t, \quad \pi b_{n} = \int_{-\pi}^{\pi}\! f(t) \sin nt \, d t, \] the series \( \frac{1}{2} a_{\:\!0} + \sum_{\:\!n=1}^{\:\!\infty} \left( a_{n} \cos nz + b_{n} \sin nz \right), \) converges to the sum \(\frac{1}{2} \left\{ f(x+0) + f(x-0) \right\}\).[6]
Later, Riemann and Cantor developed the theory of trigonometrical series generally, while still more recently Hurwitz, Fejér and others have investigated properties of Fourier series when the series does not necessarily converge. Thus Fejér has proved the remarkable theorem that a Fourier series (even if not convergent) is ‘summable \((C\:\! 1)\)’ at all points at which \(f(x \pm 0)\) exist, and its sum \((C\:\! 1)\) is \(\frac{1}{2} \left\{ f(x+0) + f(x-0) \right\}\), provided that \(\int_{\!-\pi}^{\!\:\pi} f(t) \, d t\) is an absolutely convergent integral. One of the investigations of the convergence of Fourier series which we shall give later (§9.42) is based on this result.
For a fuller account of investigations subsequent to Riemann, the reader is referred to Hobson’s Functions of a Real Variable, and to de la Vallée Poussin’s Cours d’Analyse Infinitésimale.
9.11 Nature of the region within which a trigonometrical series converges.
Consider the series \[ \frac{1}{2} a_{\:\!0} + \sum_{n=1}^{\infty} \left( a_{n} \cos nz + b_{n} \sin nz \right), \] where \(z\) may be complex. If we write \(e^{iz} = \zeta\), the series becomes \[ \frac{1}{2} a_{0} + \sum_{n=1}^{\infty} \left\{ \frac{1}{2} (a_{n} - i b_{n}) \zeta^{n} + \frac{1}{2} (a_{n} + i b_{n}) \zeta^{-n} \right\} \] This Laurent series will converge, if it converges at all, in a region in which \(a \leq \left| \, \zeta \, \right| \leq b\), where \(a,b\) are positive constants.
But, if \(z = x + iy\), \(\left| \, \zeta \, \right| = e^{-y}\), and so we get, as the region of convergence of the trigonometrical series, the strip in the \(z\) plane defined by the inequality \[ \log a \leq -y \leq \log b. \]
The case which is of the greatest importance in practice is that in which \(a = b = 1\), and the strip consists of a single line, namely the real axis.
Example 1. Let \[ f(z) = \sin z - \frac{1}{2} \sin 2z + \frac{1}{3} \sin 3z - \frac{1}{4} \sin 4z + \ldots,\] where \(z = x + iy\).
Writing this in the form \[ f(z) = - \frac{1}{2} i \left( e^{iz} - \frac{1}{2} e^{2iz} + \frac{1}{3} e^{3iz} - \cdots \right) + \frac{1}{2} i \left( e^{-iz} - \frac{1}{2} e^{-2iz} + \frac{1}{3} e^{-3iz} - \cdots \right) \] we notice that the first series converges only if \(y \geq 0\), and the second only if \(y \leq 0\).[7]
Writing \(x\) in place of \(z\) (\(x\) being real), we see that by Abel’s theorem (§3.71), \[ \begin{align*} f(x) & = \lim_{r \rightarrow 1} \left( r \sin x - \frac{1}{2} r^{2} \sin 2x + \frac{1}{3} r^{3} \sin 3x - \cdots \right) \\ & = \lim_{r \rightarrow 1} \left\{ - \frac{1}{2} i \left( r e^{ix} - \frac{1}{2} r^{2} e^{2ix} + \frac{1}{3} r^{3} e^{3ix} - \cdots \right) \right. \\ & \qquad \qquad \left. + \frac{1}{2} i \left( r e^{-ix} - \frac{1}{2} r^{2} e^{-2ix} + \frac{1}{3} r^{3} e^{-3ix} - \cdots \right) \right\} \end{align*} \]
This is the limit of one of the values of \[ - \frac{1}{2} i \log (1 + r e^{ix}) + \frac{1}{2} i \log (1 + r e^{-ix}), \] and as \(r \rightarrow 1\) (if \(-\pi < x < \pi\)), this tends to \(\frac{1}{2} x + k\pi\), where \(k\) is some integer.
Now \(\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n-1} \sin nx}{n}\) converges uniformly (§3.35 example 1) and is therefore continuous in the range \(-\pi + \delta \leq x \leq \pi - \delta\), where \(\delta\) is any positive constant.
Since \(\frac{1}{2} x\) is continuous, \(k\) has the same value wherever \(x\) lies in the range; and putting \(x=0\), we see that \(k=0\).
Therefore, when \(-\pi < x < \pi\), \[ f(x) = \frac{1}{2} x. \]
But, when \(\pi < x < 3\pi\), \[ f(x) = f(x - 2\pi) = \frac{1}{2} (x - 2\pi) = \frac{1}{2} x - \pi, \] and generally, if \((2n - 1) \pi < x < (2n + 1) \pi\), \[ f(x) = \frac{1}{2} x - n \pi. \]
We have thus arrived at an example in which \(f(x)\) is not represented by a single analytical expression.
It must be observed that this phenomenon can only occur when the strip in which the Fourier series converges is a single line. For if the strip is not of zero breadth, the associated Laurent series converges in an annulus of non-zero breadth and represents an analytic function of \(\zeta\) in that annulus; and, since \(\zeta\) is an analytic function of \(z\), the Fourier series represents an analytic function of \(z\);[8] such a series might be \[ r \sin x - \frac{1}{2} r^{2} \sin 2x + \frac{1}{3} r^{3} \sin 3x - \cdots, \] where \(0 < r < 1\); its sum is \(\arctan \left(\dfrac{r \sin x}{1 + r \cos x}\right)\), the \(\arctan\) always representing an angle between \(\pm \frac{1}{2} \pi\).[9]
Example 2. When \(-\pi \leq x \leq \pi\),
\[ \sum_{n=1}^{\infty} \frac{(-)^{n-1} \cos nx}{n^{2}} = \frac{1}{12} \pi^{2} - \frac{1}{4} x^{2}. \]
The series converges only when \(x\) is real; by §3.34 the convergence is then absolute and uniform.
Since \[ \frac{1}{2} x = \sin x - \frac{1}{2} \sin 2x + \frac{1}{3} \sin 3x - \cdots \quad (-\pi + \delta \leq x \pi - \delta, \delta > 0), \] and this series converges uniformly, we may integrate term-by-term from \(0\) to \(x\) (§4.7), and consequently \[ \frac{1}{4} x^{2} = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} (1 - \cos nx)}{n^{2}} \quad (-\pi + \delta \leq x \leq \pi - \delta). \]
That is to say, when \(-\pi + \delta \leq x \leq \pi - \delta\), \[ C - \frac{1}{4} x^{2} = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} \cos nx}{n^{2}}, \] where \(C\) is a constant, at present undetermined.
But since the series on the right converges uniformly throughout the range \(-\pi \leq x \leq \pi\), its sum is a continuous function of \(x\) in this extended range; and so, proceeding to the limit when \(x \rightarrow \pm \pi\), we see that the last equation is still true when \(x = \pm \pi\).
To determine \(C\), integrate each side of the equation (§4.7) between the limits \(-\pi, \:\!\pi\); and we get \[ 2 \pi C - \frac{1}{6} \pi^{3} = 0. \]
Consequently \[ \frac{1}{12} \pi^{2} - \frac{1}{4} x^{2} = \sum_{n=1}^{\infty} \frac{ (-1)^{n-1} \cos nx }{ n^{2} } \quad (-\pi \leq x \leq \pi). \]
Example 3. By writing \(\pi - 2x\) for \(x\) in example 2, shew that \[ \sum_{n=1}^{\infty} \frac{\sin^{2} nx}{n^{2}} = \begin{cases} \frac{1}{2} x (\pi - x) & 0 \leq x \leq \pi, \\ \frac{1}{2} \left\{ \pi \left| \, x \, \right| - x^{2}\right\} & -\pi \leq x \leq \pi. \end{cases} \]
9.12 Values of the Coefficients in Terms of the Sum of a Trigonometrical Series.
Let the trigonometrical series \( \displaystyle \frac{1}{2} c_{\:\!0} + \sum_{n=1}^{\infty} (c_{n} \cos nx + d_{n} \sin nx) \) be uniformly convergent in the range \((-\pi, \pi)\) and let its sum be \(f(x)\). Using the obvious results \[ \begin{align*} \int_{-\pi}^{\pi}\! \cos mx \cos nx \, d x =& \begin{cases} 0 & m \neq n \\ \pi & m = n \neq 0, \end{cases} \\ \int_{-\pi}^{\pi}\! \sin mx \sin nx \, d x =& \begin{cases} 0 & m \neq n \\ \pi & m = n \neq 0, \end{cases} \\ \int_{-\pi}^{\pi}\! \, d x =& 2\pi, \end{align*} \] we find, on multiplying the equation \( \displaystyle \frac{1}{2} c_{\:\!0} + \sum_{n=1}^{\infty} (c_{n} \cos nx + d_{n} \sin nx) = f(x) \) by \(\cos nx\) or by \(\sin nx\)[10] and integrating term-by-term (§4.7),[11] \[ \pi c_{n} = \int_{-\pi}^{\pi}\! f(x) \cos nx \, d x, \quad \pi d_{n} = \int_{-\pi}^{\pi}\! f(x) \sin nx \, d x. \]
Corollary. A trigonometrical series uniformly convergent in the range \((-\pi, \pi)\) is a Fourier series.
Note. Lebesgue has given a proof (Séries trigonométriques, p. 124) of a theorem communicated to him by Fatou that the trigonometrical series \( \left.\sum_{\:\!n=2}^{\:\!\infty} \sin nx \middle/ \!\log n\right.\), which converges for all real values of \(x\) (§2.31 example 1), is not a Fourier series.