05 A first look at Fourier series, part 1

The fundamental question

One of the fundamental questions raised when solving an initial-value problem via separation of variables is:

The Fourier question

Given a function \(f:\mathbb{R} \to \mathbb{R}\), when can we write it as a series of the form

\[ f(\theta) = \frac{a_0}{2} + \sum_{n=1}^\infty \left( a_n \cos{n\theta} + b_n \sin{n\theta}\right)? \]

When does the series on the right converge?
If it does converge, does it converge to \(f\)?
What are the coefficients \(a_n\) and \(b_n\)?

Note we use the variable \(\theta\), rather than \(x\). No harm here.
If the answer to (2) is always, then the function \(f\) had better be periodic with period \(2\pi\), so we will focus always on these types of functions.

Exericse 1: Bringing in complex numbers

Recall that from Euler’s formula \(e^{i\theta} = \cos{\theta} + i\sin{\theta}\), we can express the sine and cosine functions in terms of complex exponentials as follows:

\[ \cos{\theta} = \frac{e^{i\theta} + e^{-i\theta}}{2}, \quad \sin{\theta} = \frac{e^{i\theta} - e^{-i\theta}}{2i}. \]

Express the series \[ \frac{a_0}{2} + \sum_{n=1}^\infty \left( a_n \cos{n\theta} + b_n \sin{n\theta}\right) \] in terms of complex exponentials with coefficients \(c_n\) for \(n\in \mathbb{Z}\).
Conversely, starting from the series of complex exponentials that you found in part (1), express it back in terms of sines and cosines with coefficients \(a_n\) and \(b_n\) for \(n\geq 0\).

Deriving a formula for \(c_n\)

Now, suppose we take for granted that \[ f(\theta) = \sum_{n=-\infty}^\infty c_n e^{in\theta}, \tag{1} \] and that it is permissible to integrate term-by-term. (These are assumptions, not claims of fact!)
How might we find a formula for the coefficients \(c_n\)?
Integrate both sides of (1) against \(e^{-ik\theta}\) for some fixed integer \(k\), from \(-\pi\) to \(\pi\): \[ \int_{-\pi}^\pi f(\theta) e^{-ik\theta} \, d\theta = \sum_{n=-\infty}^\infty c_n \int_{-\pi}^\pi e^{i(n-k)\theta} \, d\theta. \]
Now, compute the integrals on the right-hand side: \[ \int_{-\pi}^\pi e^{i(n-k)\theta} \, d\theta = \begin{cases} 2\pi & : n=k, \\ 0 & : n \neq k. \end{cases} \]
Therefore, we have: \[ \int_{-\pi}^\pi f(\theta) e^{-ik\theta} \, d\theta = 2\pi c_k, \] which is the same, after relabelling, as the formula \[ c_n = \frac{1}{2\pi} \int_{-\pi}^\pi f(\theta) e^{-in\theta} \, d\theta. \]

Deriving formulas for \(a_n\) and \(b_n\)

From Exercise 1, we know that \[ a_0 = 2c_0, \quad a_n = c_n + c_{-n}, \quad b_n = i(c_n - c_{-n}) \] for \(n\geq 1\).
Therefore, we can express \(a_n\) and \(b_n\) in terms of integrals of \(f\) as follows: \[ a_0 = \frac{1}{\pi} \int_{-\pi}^\pi f(\theta) \, d\theta, \quad a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(\theta) \cos{n\theta} \, d\theta, \quad b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(\theta) \sin{n\theta} \, d\theta, \] for \(n\geq 1\).

Definition

Let \(f:\mathbb{R} \to \mathbb{R}\) be an integrable function with period \(2\pi\).

The numbers \(c_n\) for \(n\in \mathbb{Z}\), \(a_n\) for \(n\geq 0\), and \(b_n\) for \(n\geq 1\), defined by \[ c_n = \frac{1}{2\pi} \int_{-\pi}^\pi f(\theta) e^{-in\theta} \, d\theta, \quad a_n = \frac{1}{\pi} \int_{-\pi}^\pi f(\theta) \cos{n\theta} \, d\theta, \quad b_n = \frac{1}{\pi} \int_{-\pi}^\pi f(\theta) \sin{n\theta} \, d\theta, \] are called the Fourier coefficients of \(f\).
The series \[ \sum_{n=-\infty}^\infty c_n e^{in\theta} \quad \text{or} \quad \frac{a_0}{2} + \sum_{n=1}^\infty \left( a_n \cos{n\theta} + b_n \sin{n\theta}\right) \] is called the Fourier series of \(f\).

WARNING!!!

Nothing has been claimed about the convergence of the Fourier series!

Exercise 2: Fourier coefficients of even and odd functions

Let \(f:\mathbb{R} \to \mathbb{R}\) be an integrable function with period \(2\pi\).

If \(f\) is even, show that \(b_n=0\) for all \(n\geq 1\), and that \[ a_n = \frac{2}{\pi} \int_0^\pi f(\theta) \cos{n\theta} \, d\theta. \]
If \(f\) is odd, show that \(a_n=0\) for all \(n\geq 0\), and that \[ b_n = \frac{2}{\pi} \int_0^\pi f(\theta) \sin{n\theta} \, d\theta. \]

Exercise 3: Computing a Fourier series

Define the function \(f(\theta) = |\theta|\) for \(\theta \in [-\pi, \pi]\), and extend it to a function on \(\mathbb{R}\) with period \(2\pi\). Compute the Fourier coefficients and series of \(f\).

import numpy as np
import matplotlib.pyplot as plt

blue = "#3399FF"
white = "#FAF9F6"
theta_range = np.linspace(-4 * np.pi, 4 * np.pi, 400)


def f(theta):
    theta_wrapped = ((theta + np.pi) % (2 * np.pi)) - np.pi
    return np.abs(theta_wrapped)


def f_n(k, theta):
    return np.cos((2 * k - 1) * theta) / (2 * k - 1) ** 2


def f_approx(theta, n):
    return np.pi / 2 - (4 / np.pi) * sum(f_n(k, theta) for k in range(1, n + 1))


fig, axes = plt.subplots(nrows=2, ncols=2, sharex=True, sharey=True)
fig.set_facecolor(white)


axes[0, 0].plot(theta_range, f(theta_range), color=blue)
axes[0, 0].set_title(r"$f(\theta) = |\theta|$ extended periodically")

n = 1
axes[0, 1].plot(theta_range, f_approx(theta_range, n), color=blue)
axes[0, 1].set_title(r"$S_1(\theta)$")

n = 2
axes[1, 0].plot(theta_range, f_approx(theta_range, n), color=blue)
axes[1, 0].set_title(r"$S_2(\theta)$")

n = 5
axes[1, 1].plot(theta_range, f_approx(theta_range, n), color=blue)
axes[1, 1].set_title(r"$S_5(\theta)$")

for ax in axes.flatten():
    ax.set_facecolor(white)

plt.tight_layout()
plt.show()