Fourier Series and the First Heat Equation

In 1807 Joseph Fourier submitted a memoir to the Paris Academy claiming that the temperature in a solid could be found by writing the initial temperature as a sum of sines and cosines, and letting each one decay at its own rate. The claim that an arbitrary function could be written that way was greeted with scepticism, and making sense of it occupied analysts for a century: it is one of the main reasons the concepts of function, convergence and integral in Books 2A and 2B were made precise at all. Fourier's book, Théorie analytique de la chaleur, appeared in 1822.

This chapter does Fourier's analysis in the setting of periodic functions, and then solves Fourier's problem: heat flow on a ring. Here the heat equation, the equation behind thread H of this guidebook and, eventually, behind Ricci flow, is solved completely and explicitly. Every property of heat flow that later chapters prove with harder tools can be seen here in the formula: high frequencies die fastest, so the solution becomes smooth instantly; an energy decreases; the maximum can only fall; and the solution is an average of the initial data against a kernel. When Hamilton described Ricci flow as a heat equation for the metric (11A.1 The Equation and Its First Solutions), these are the properties he meant.

By the end of this chapter you will be able to:

• compute Fourier coefficients, and use orthonormality of $e^{2\pi ikx}$ to find best approximations;
• explain why Fourier series of continuous functions need not converge, and prove that their Fejér means do;
• prove Plancherel's identity and use it to sum series such as $\sum 1/k^2$;
• solve the heat equation on a ring by Fourier series, and prove instant smoothing, energy decay, the maximum principle and exponential convergence to the mean;
• explain the Gibbs phenomenon, and why positive kernels avoid it.

7.1Seasons underground

The decisive feature is that each frequency decays at its own rate, faster for higher frequencies: the damping depth is proportional to $1/\sqrt\omega$. On a ring of metal, where heat flows around instead of down, the same principle gives the complete solution of the heat equation. To use it we need to break an arbitrary periodic function into frequencies.

7.2Periodic functions and their coefficients

Work with functions of period $1$: $f(x + 1) = f(x)$. Equivalently, functions on the circle $\mathbb{R}/\mathbb{Z}$, the real line with $x$ and $x + 1$ identified (2A.2 Sets, Functions and Equivalence). Let $C(\mathbb{R}/\mathbb{Z})$ be the continuous periodic functions with complex values, a complete metric space with the sup metric (2B.5 Uniform Convergence and Arzelà–Ascoli).

This behaves like the dot product on $\mathbb{C}^n$: it is linear in the first slot, conjugate-symmetric, and $\langle f, f\rangle > 0$ unless $f = 0$ (by the positivity argument of 2B.1 Metric Spaces). The Cauchy–Schwarz inequality $|\langle f, g\rangle| \leq \|f\|_2\|g\|_2$ and the triangle inequality for $\|\cdot\|_2$ follow exactly as in 2B.1 Metric Spaces. So $d_2(f, g) = \|f - g\|_2$ is a metric, the root-mean-square distance.

The building blocks are the characters $e_k(x) = e^{2\pi ikx}$, $k \in \mathbb{Z}$, which go $|k|$ times around the unit circle as $x$ runs over $[0, 1]$.

A trigonometric polynomial is a finite sum $\sum_{|k| \leq N} c_ke_k$. Its coefficients can be recovered by taking inner products: $c_k = \langle P, e_k\rangle$. For a general $f$ this suggests a definition.

For real $f$, $\hat f(-k) = \overline{\hat f(k)}$, and the terms pair up into real cosines and sines: $\hat f(k)e_k + \hat f(-k)e_{-k} = 2\operatorname{Re}\big(\hat f(k)e^{2\pi ikx}\big)$.

The partial sum $S_Nf$ is the best approximation to $f$ by trigonometric polynomials of degree at most $N$, in the root-mean-square sense. This is Pythagoras: $f - S_Nf$ is orthogonal to every $e_k$ with $|k| \leq N$, so for any trigonometric polynomial $P$ of degree $\leq N$,

Taking $P = 0$ gives Bessel's inequality: $\sum_{|k|\leq N}|\hat f(k)|^2 = \|S_Nf\|_2^2 \leq \|f\|_2^2$ for every $N$. In particular $\hat f(k) \to 0$ as $|k| \to \infty$.

7.3Does the Fourier series converge?

The natural hope is that $S_Nf \to f$. Write the partial sum as an average:

Expressions of this kind are periodic convolutions, $(f * g)(x) = \int_0^1 f(y)g(x - y)\,dy$, and $S_Nf = f * D_N$. If $D_N$ were an approximate identity in the sense of 2B.5 Uniform Convergence and Arzelà–Ascoli, non-negative with integral $1$ and concentrating at $0$, we would have $S_Nf \to f$ uniformly. It has integral $1$ and its peak at $0$ grows. But it is not non-negative: it oscillates, with side lobes that decay slowly (Figure 7.1). Its total absolute mass $\int_0^1|D_N|$ grows like $\log N$, and that is enough to wreck convergence: there are continuous functions whose Fourier series diverge at a point (du Bois-Reymond, 1873).

The cure is to average the partial sums.

The last formula (Exercise 7.10) is the important one: $F_N \geq 0$. Averaging has cancelled the oscillation.

Three consequences follow, and they are what make Fourier series usable.

How fast $\hat f(k)$ decays measures how smooth $f$ is. Integrating by parts (2A.11 The Riemann Integral, with no boundary terms because $f$ is periodic), $\widehat{f'}(k) = 2\pi ik\,\hat f(k)$. So if $f$ is $C^2$, then $|\hat f(k)| = \frac{|\widehat{f''}(k)|}{4\pi^2k^2} \leq \frac{\sup|f''|}{4\pi^2k^2}$, which is summable, and the Fourier series converges uniformly. Smoothness of $f$ is decay of $\hat f$. That sentence, made quantitative, is the theory of Sobolev spaces (4A.9 Sobolev Spaces).

7.4Fourier's solution of the heat equation on a ring

Now Fourier's problem. A thin ring of metal of circumference $1$ has temperature $u(x, t)$ at position $x \in \mathbb{R}/\mathbb{Z}$ and time $t$. Choosing units so the diffusivity is $1$,

Separate the frequencies. Try $u(x, t) = \sum_k a_k(t)e^{2\pi ikx}$. Since $\partial_x^2e_k = -4\pi^2k^2e_k$, the equation decouples into one ordinary differential equation per frequency:

by the uniqueness in 2B.6 Power Series, Exponentials and Bump Functions. So the candidate solution is

In words: each frequency decays exponentially, at a rate proportional to the square of the frequency. The $k = 0$ term, the mean temperature, never decays. Frequency $1$ decays like $e^{-39.5t}$; frequency $10$ like $e^{-3948t}$.

7.4.1What the formula shows

Each property below is visible in $(*)$, and each is a theme that later books prove for much harder equations, where no formula is available.

1. Instant smoothing. However rough $f$ is (here, merely continuous; in Course 3, merely square-integrable), $u(\cdot, t)$ is infinitely differentiable for every $t > 0$, because $e^{-4\pi^2k^2t}$ crushes the high frequencies. The heat equation cannot be run backwards in general: running it backwards would multiply $\hat f(k)$ by $e^{+4\pi^2k^2t}$, and only very special $f$ survive that (Exercise 7.12). Smoothing is the reason Ricci flow is useful at all: it improves a metric, and the improvement is quantified by Shi's derivative estimates (11A.3 Short-Time Existence and Uniqueness), the analogue of the bound in part 1 of the proof.

2. Energy decreases. By Plancherel, $\int_0^1|u(x, t)|^2dx = \sum_k|\hat f(k)|^2e^{-8\pi^2k^2t}$, which visibly decreases in $t$. Directly, as in the rehearsal of 2A.11 The Riemann Integral, integrating by parts with no boundary terms,

Likewise the Dirichlet energy $\int(\partial_xu)^2 = \sum 4\pi^2k^2|\hat f(k)|^2e^{-8\pi^2k^2t}$ decreases. In fact the heat equation is the direction of steepest descent of the Dirichlet energy, the gradient flow of $\tfrac12\int|\partial_xu|^2$ (Exercise 7.15). This is thread V's first appearance: Ricci flow is, after Perelman's modification, the gradient flow of his $\mathcal{F}$-functional (12A.2 Ricci Flow as a Gradient Flow).

3. Convergence to equilibrium, at a rate set by the first frequency. Let $\bar f = \hat f(0)$ be the mean. Then

The temperature evens out exponentially fast, at the rate $4\pi^2$ of the lowest non-zero frequency. This spectral gap is equivalent to an inequality of Poincaré type (Exercise 7.14), and it reappears as the first eigenvalue of the Laplacian on a manifold (9B.7 The Heat Equation on a Manifold).

4. The maximum principle. Since $u(\cdot, t) = f * H_t$ with $H_t \geq 0$ and $\int H_t = 1$, each value $u(x, t)$ is an average of values of $f$. So $\min f \leq u(x, t) \leq \max f$: the heat equation creates no new maxima or minima, and in fact $\max_x u(x, t)$ is non-increasing in $t$ (apply the same argument starting from any time $s$). For the heat equation on a ring this follows from the positivity of a kernel; in 6A.4 Maximum Principles it is proved for general heat equations by the second-derivative test (2A.10 Derivatives, 2B.8 Calculus in Several Variables), and in 11A.4 Maximum Principles under Ricci Flow Hamilton extends it to tensors.

7.5The Gibbs phenomenon

Fourier series of functions with jumps converge badly near the jumps. For the square wave ($+1$ on $(0, \tfrac12)$, $-1$ on $(\tfrac12, 1)$), the partial sums overshoot the jump by a fixed amount that does not shrink as $N$ grows; the overshoot just moves closer to the jump (Figure 7.3). In the limit, the partial sums rise to $\frac2\pi\int_0^\pi\frac{\sin t}{t}\,dt \approx 1.179$, overshooting the value $1$ by about $9\%$ of the jump of size $2$.

The overshoot is caused by the negative lobes of the Dirichlet kernel. The Fejér means, which average against a positive kernel, never overshoot: by the argument of property 4, they stay between $-1$ and $1$. Positivity of the averaging kernel is what makes both the maximum principle and the absence of overshoot work.

7.6History

Fourier's 1807 memoir was judged by Lagrange, Laplace, Monge and Lacroix; its prize-winning revision (1811) and the 1822 book established the method, and Chapter IV of the book treats the movement of heat in a ring. Peter Gustav Lejeune Dirichlet gave the first rigorous convergence theorem in 1829, for piecewise monotone functions. Paul du Bois-Reymond constructed a continuous function whose Fourier series diverges at a point in 1873. In 1900, aged twenty, Lipót Fejér proved that the Cesàro means of the Fourier series of every continuous function converge uniformly. The overshoot at jumps was described by Henry Wilbraham in 1848 and again by J. Willard Gibbs in 1899, whose name it carries. The questions Fourier raised, what a function is and in what sense a series converges, led to Riemann's integral (1854), to Cantor's set theory (which began with sets of uniqueness for trigonometric series), and to Lebesgue's integral (1902), the subject of Course 3.

7.7Exercises

Solution

$\|g\|_2^2 = \int_0^1(x - \tfrac12)^2dx = \frac1{12}$, and $\sum_{k\neq0}|\hat g(k)|^2 = 2\sum_{k\geq1}\frac{1}{4\pi^2k^2}$. Equate: $\sum\frac1{k^2} = \frac{4\pi^2}{2}\cdot\frac{1}{12} = \frac{\pi^2}6$.

Solution

(a) With $v = e^{i\omega t - (1+i)z/d}$: $v_t = i\omega v$ and $v_{zz} = \frac{(1+i)^2}{d^2}v = \frac{2i}{d^2}v$, so $v_t = \kappa v_{zz}$ exactly when $\omega = 2\kappa/d^2$. Take real parts. (b) Annual: $\omega = 2\pi/(3.156\times10^7\text{ s})$, $d \approx 2.24$ m. Daily: $d \approx 0.117$ m. (c) $e^{-z/d} = 0.01$ at $z = d\ln 100 \approx 10.3$ m, with a lag of $\frac{\ln 100}{2\pi} \approx 0.73$ of a year, about $8.8$ months.

Solution

The coefficients of $u(\cdot, t)$ satisfy $a_k' = -4\pi^2k^2a_k$, so $\hat f(k) = a_k(0) = a_k(-1)e^{-4\pi^2k^2}$, and $|a_k(-1)| \leq \sup|u(\cdot, -1)|$. A function with $\hat f(k) = 1/k^2$ ($k \neq 0$) is continuous by the M-test, but $1/k^2$ is not $O(e^{-4\pi^2k^2})$.

Hint

Suppose $u_\varepsilon = u + \varepsilon(1 + t)$ first becomes $0$ at a time $t_1 > 0$ and point $x_1$. There $\partial_tu_\varepsilon \leq 0$ and $\partial_x^2u_\varepsilon \geq 0$, but $\partial_tu_\varepsilon - \partial_x^2u_\varepsilon = \varepsilon > 0$. For the last step, $(f * H_t)(x) = \int f(y)H_t(x - y)dy$, and if $H_t(x_0) < 0$ a bump concentrated near $0$ makes this negative at $x = x_0$.

Solution

(a) $\int|u - \bar u|^2 = \sum_{k\neq0}|\hat u(k)|^2 \leq \sum_{k\neq0}\frac{4\pi^2k^2}{4\pi^2}|\hat u(k)|^2 = \frac{1}{4\pi^2}\int|u'|^2$. Equality iff $\hat u(k) = 0$ for $|k| \geq 2$: $u = \bar u + a\cos 2\pi x + b\sin 2\pi x$. (b) The mean is constant (2A.11 The Riemann Integral), so $E' = 2\int(u - \bar u)u_t = 2\int(u - \bar u)u_{xx} = -2\int u_x^2 \leq -8\pi^2E$. Then $(e^{8\pi^2t}E)' \leq 0$.