7.2 Fourier series
For an integrable function \(f\) on \([-\pi , \pi ]\), the Fourier coefficients and the partial sums of the Fourier series are \begin {gather*} a_k = \frac 1\pi \int _{-\pi }^\pi f(x)\cos kx\,dx, \qquad b_k = \frac 1\pi \int _{-\pi }^\pi f(x)\sin kx\,dx, \\ S_Nf(x) = \frac {a_0}2 + \sum _{k = 1}^N\bigl (a_k\cos kx + b_k\sin kx\bigr ) . \end {gather*} The functions \(1, \cos kx, \sin kx\) are orthogonal on \([-\pi , \pi ]\), with \(\int \cos ^2kx = \int \sin ^2kx = \pi \) for \(k \geq 1\). An odd function has only sine terms, an even one only cosines; the sine series of a function on \([0, \pi ]\) is the Fourier series of its odd extension, which is how (7.1) arises (with \(x\) rescaled). A function on an interval of length \(2L\) is handled by the substitution \(x \mapsto \pi x/L\).
Lemma 7.2.1 (Bessel’s inequality). If \(f^2\) is integrable on \([-\pi , \pi ]\), then \(\frac {a_0^2}2 + \sum _{k = 1}^\infty (a_k^2 + b_k^2) \leq \frac 1\pi \int _{-\pi }^\pi f^2\,dx\). In particular \(a_k, b_k \to 0\).
Proof. By orthogonality, \(\int fS_Nf = \int (S_Nf)^2 = \pi \bigl (\frac {a_0^2}2 + \sum _1^N(a_k^2 + b_k^2)\bigr )\), so \(0 \leq \int (f - S_Nf)^2 = \int f^2 - \pi \bigl (\frac {a_0^2}2 + \sum _1^N(a_k^2 + b_k^2)\bigr )\). Let \(N \to \infty \). □
The same computation shows that \(S_Nf\) is the best approximation to \(f\) in the mean-square sense among all trigonometric polynomials of degree \(N\): adding anything orthogonal to \(S_Nf - f\) only increases the error. The statement that the coefficients tend to zero, for any integrable function, is the Riemann–Lebesgue lemma.
Theorem 7.2.2 (Dirichlet). Let \(f\) be \(2\pi \)-periodic and piecewise \(C^1\) (finitely many pieces in each period, each \(C^1\) up to its ends). Then for every \(x\), \[ S_Nf(x) \to \tfrac 12\bigl (f(x+) + f(x-)\bigr ) , \] which is \(f(x)\) wherever \(f\) is continuous.
Proof. Substituting the coefficients into \(S_Nf\) and summing the geometric series \(\frac 12 + \sum _1^N\cos ky = \sin ((N + \frac 12)y)/(2\sin \frac y2)\) gives \[ S_Nf(x) = \int _{-\pi }^\pi D_N(y)f(x + y)\,dy, \qquad D_N(y) = \frac {\sin \bigl ((N + \frac 12)y\bigr )}{2\pi \sin \frac y2} , \] the Dirichlet kernel, which is even and has \(\int _0^\pi D_N = \frac 12\) (integrate the cosine sum). Hence \[ S_Nf(x) - \tfrac 12\bigl (f(x+) + f(x-)\bigr ) = \int _0^\pi \sin \bigl ((N + \tfrac 12)y\bigr )\,\bigl (q_+(y) + q_-(y)\bigr )\,dy , \] where \[ q_\pm (y) = \frac {f(x \pm y) - f(x\pm )}{2\pi \sin \frac y2} . \] Since \(f\) has one-sided derivatives at \(x\), \(q_\pm (y)\) tends to a finite limit as \(y \to 0+\), so \(q_\pm \) is bounded and piecewise continuous on \((0, \pi ]\). The functions \(\sin ((N + \frac 12)y)\) are orthogonal on \([0, \pi ]\), each with \(\int _0^\pi \sin ^2 = \frac \pi 2\), so by Bessel’s inequality for this system the integrals, which are \(\frac \pi 2\) times the coefficients of \(q_+ + q_-\), tend to \(0\). □
The proof shows that convergence at \(x\) depends only on the behaviour of \(f\) near \(x\): the localisation principle. Mere continuity is not enough: du Bois-Reymond (1873) constructed a continuous function whose Fourier series diverges at a point. In the other direction, Carleson (1966) proved that the Fourier series of every square-integrable function converges almost everywhere.
Theorem 7.2.3 (Uniform convergence). If \(f\) is \(2\pi \)-periodic, continuous and piecewise \(C^1\), then \(\sum (|a_k| + |b_k|) < \infty \), and \(S_Nf \to f\) uniformly.
Proof. Integrating by parts on each piece, the boundary terms cancel because \(f\) is continuous and periodic, and the coefficients of \(f'\) are \(a_k' = kb_k\), \(b_k' = -ka_k\). By the Cauchy–Schwarz inequality and Bessel’s inequality for \(f'\), \[ \sum _{k \geq 1}\bigl (|a_k| + |b_k|\bigr ) = \sum _{k \geq 1}\frac {|b_k'| + |a_k'|}k \leq \Bigl (2\sum _{k \geq 1}(a_k'^2 + b_k'^2)\Bigr )^{1/2}\Bigl (\sum _{k \geq 1}\frac 1{k^2}\Bigr )^{1/2} < \infty . \] By the Weierstrass M-test the series converges uniformly, and by Theorem 7.2.2 its sum is \(f\). □
The argument shows the general principle: smoothness of \(f\) is decay of its coefficients. If \(f\) and its first \(m - 1\) derivatives are continuous and periodic and \(f^{(m)}\) is square-integrable, then \(\sum k^{2m}(a_k^2 + b_k^2) < \infty \); a jump in \(f\) itself allows only decay like \(1/k\), a jump in \(f'\) like \(1/k^2\).
Theorem 7.2.4 (Parseval). If \(f\) is piecewise continuous on \([-\pi , \pi ]\), then \(\int _{-\pi }^\pi (f - S_Nf)^2\,dx \to 0\), and \[ \frac 1\pi \int _{-\pi }^\pi f^2\,dx = \frac {a_0^2}2 + \sum _{k = 1}^\infty \bigl (a_k^2 + b_k^2\bigr ) . \]
Proof. Given \(\varepsilon > 0\), there is a continuous, periodic, piecewise linear \(g\) with \(\int (f - g)^2 < \varepsilon \) (round off each jump over a short interval). By Theorem 7.2.3, \(S_Ng \to g\) uniformly, so \(\int (g - S_Ng)^2 < \varepsilon \) for large \(N\). Since \(S_Nf\) is the best mean-square approximation of degree \(N\), \[ \|f - S_Nf\| \leq \|f - S_Ng\| \leq \|f - g\| + \|g - S_Ng\| < 2\sqrt \varepsilon , \] writing \(\|h\|^2 = \int h^2\). So the error tends to \(0\), and the identity is the computation in the proof of Lemma 7.2.1 with equality in the limit. □
Parseval’s identity says that the trigonometric system is complete: no non-zero function is orthogonal to all of it. For \(f(x) = x\), \(b_k = 2(-1)^{k + 1}/k\), and the identity gives \(\sum 4/k^2 = \frac 1\pi \int x^2 = \frac {2\pi ^2}3\), that is, \(\sum _{k \geq 1}1/k^2 = \pi ^2/6\) (Software 7.5.1).
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