Fourier series

Periodic functions are sums of sines and cosines; this is linear algebra in a space of functions.

1. Orthogonality of the trigonometric system (Prop. 1.1) and Fourier series (Def. 1.2), with a worked example (Example 1.3).
2. Dirichlet's convergence theorem (Thm 1.4).
3. Sine and cosine series via odd and even extensions (Prop. 1.5); complex form (Def. 1.6).
4. Parseval's identity (Thm 1.7) and the Gibbs phenomenon (Example 1.8); differentiation and decay of coefficients (Prop. 1.9).

On $[-L, L]$, for integers $m, n\ge0$:

and $\int_{-L}^L1\,\mathrm{d}x = 2L$.

For an integrable $2L$-periodic $f$, the Fourier series is

The formulas are forced: if $f$ equals such a series (with termwise integration allowed), multiplying by a basis function and integrating isolates one coefficient.

• Square wave $f = -1$ on $(-\pi, 0)$, $1$ on $(0, \pi)$: odd, $b_n = \frac2\pi\int_0^\pi\sin nx = \frac{2(1 - (-1)^n)}{n\pi}$, so $f\sim\frac4\pi\big(\sin x + \frac{\sin3x}3 + \frac{\sin5x}5 + \cdots\big)$.
• Sawtooth $f(x) = x$ on $(-\pi, \pi)$: odd, $b_n = \frac{2(-1)^{n+1}}n$, so $x\sim2\sum\frac{(-1)^{n+1}}n\sin nx$. Putting $x = \frac\pi2$ in the square wave gives $1 - \frac13 + \frac15 - \cdots = \frac\pi4$.

If $f$ is $2L$-periodic and piecewise $C^1$ (finitely many jumps, with one-sided derivatives), its Fourier series converges at every $x$ to $\frac12\big(f(x^-) + f(x^+)\big)$: to $f(x)$ where $f$ is continuous, and to the midpoint of the jump otherwise. If in addition $f$ is continuous everywhere, the convergence is uniform. (Proof via the Dirichlet kernel; omitted.)

A function on $(0, L)$ can be expanded as a sine series $\sum b_n\sin\frac{n\pi x}L$ with $b_n = \frac2L\int_0^Lf\sin\frac{n\pi x}L$ (the series of its odd $2L$-periodic extension), or as a cosine series $\frac{a_0}2 + \sum a_n\cos\frac{n\pi x}L$ with $a_n = \frac2L\int_0^Lf\cos\frac{n\pi x}L$ (even extension). The sine series converges to $0$ at $x = 0, L$; the cosine series has derivative $0$ there formally.

For $2\pi$-periodic $f$, $f\sim\sum_{n\in\mathbb{Z}}c_ne^{inx}$ with $c_n = \frac1{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}\,\mathrm{d}x$. Then $c_n = \frac{a_n - ib_n}2$ and $c_{-n} = \frac{a_n + ib_n}2$ for $n\ge1$, $c_0 = \frac{a_0}2$; for real $f$, $c_{-n} = \overline{c_n}$.

For square-integrable $f$ on $(-L, L)$,

and the partial sums converge to $f$ in mean square. The partial sum $S_N$ is the best approximation to $f$ in mean square by trigonometric polynomials of degree $\le N$ (Bessel's inequality gives $\le$; equality is completeness, quoted).

Near a jump, partial sums overshoot: for the square wave, $\max S_N\to\frac2\pi\int_0^\pi\frac{\sin t}t\,\mathrm{d}t\approx1.179$ as $N\to\infty$, an overshoot of about $9\%$ of the jump size ($2$), located at distance $\approx\frac\pi{N}$ from the jump. The overshoot never disappears; it only narrows, consistent with convergence being pointwise but not uniform.

If $f$ is $2\pi$-periodic, continuous and piecewise $C^1$, then $f'$ has Fourier coefficients $nb_n$ (for $\cos$) and $-na_n$ (for $\sin$); i.e. termwise differentiation gives the series of $f'$. Consequently smoother $f$ has faster-decaying coefficients: jumps give $O(\frac1n)$, kinks $O(\frac1{n^2})$, and $f\in C^k$ gives $o(n^{-k})$.

• Use symmetry first: odd ⇒ sines only, even ⇒ cosines only.
• Integrate by parts; $\cos n\pi = (-1)^n$, $\sin n\pi = 0$.
• Check decay: jumps $\sim\frac1n$, kinks $\sim\frac1{n^2}$.
• Check a value: the series at a continuity point must give $f(x)$; at a jump, the midpoint.