Fourier Analysis: Introduction

① What are we trying to do?

The aim of a Fourier series is to express a complicated function as a sum of simple $\sin$ and $\cos$ waves.

② Representing a periodic waveform as a sum of $\sin$ and $\cos$

Assume that the complicated waveform $f(x)$ has period $2\pi$.

$$f(x + 2\pi) = f(x)$$

When we hear "period $2\pi$", the high-school $\sin$ and $\cos$ come to mind.

Remarkably, (with a few exceptions) any complicated waveform can be expressed as a sum of $\sin$ and $\cos$ as follows.

$$f(x) = \dfrac{a_0}{2} + \sum_{n=1}^{\infty}\left\{ a_n \cos(nx) + b_n \sin(nx) \right\}$$

This is called the Fourier series expansion of $f(x)$, and the coefficients $a_n$ and $b_n$ are computed by the following definite integrals.

$$\begin{align*} a_0 &= \dfrac{1}{\pi} \int_{-\pi}^{\pi} f(x)\,dx \\ a_n &= \dfrac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx)\,dx \\ b_n &= \dfrac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx)\,dx \end{align*}$$

③ Why does it matter?

After entering university: It is an important subject taught in nearly every science and engineering department — physics, electrical engineering, mechanical engineering, information science, and so on. The Fourier series studied here serves as an important foundation for more general mathematical analysis and a wide range of applications.

④ Let us try: the Fourier series expansion of $f(x)=x$

Applying the integration-by-parts formula taught in high school,

$$\int_a^b u(x)v'(x)\,dx = \left[ u(x)v(x) \right]_a^b - \int_a^b u'(x)v(x)\,dx,$$

with

$$\begin{align*} u(x) &= x, & v'(x) &= \sin(nx), \\ u'(x) &= 1, & v(x) &= -\dfrac{1}{n}\cos(nx), \end{align*}$$

we obtain the following.

\begin{equation} b_n = \frac{1}{\pi} \left[ -\frac{x}{n}\cos(nx) \right]_{-\pi}^{\pi} + \frac{1}{\pi n} \int_{-\pi}^{\pi} \cos(nx)\,dx \label{eq:bn_partial} \end{equation}

The first term of equation $\eqref{eq:bn_partial}$ becomes

$$\begin{align*} -\dfrac{1}{\cancel{\pi} n} \left[ \cancel{\pi} \cos(n\pi) - (-\cancel{\pi}) \cos(-n\pi) \right] &= -\dfrac{1}{n} \left[ \cos(n\pi) + \cos(-n\pi) \right] \\ &\quad \text{since } \cos(-n\pi) = \cos(n\pi), \\ &= -\dfrac{2\cos(n\pi)}{n} \\ &\quad \text{and since } \cos(n\pi) = (-1)^n, \\ &= \dfrac{2}{n}(-1)^{n+1}. \end{align*}$$

The second term of equation $\eqref{eq:bn_partial}$ is the integral of $\cos$ over one full period, so

Therefore $b_n$ is

$$b_n = \dfrac{2}{n}(-1)^{n+1}.$$

Since $a_0 = a_n = 0$, the function $f(x) = x$ involves only $\sin(nx)$, and its Fourier series expansion is

$$ f(x) = 2\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{n} \sin(nx).$$

⑤ Partial sums and convergence

The sum of the first $N$ terms is called the $N$th partial sum. The partial sums for $N=1, 10, 100$ are shown below.

$$ f_N(x) = 2\sum_{n=1}^{N} \dfrac{(-1)^{n+1}}{n} \sin(nx)$$

A few things are worth noticing from the graph.

1. As $N$ grows, the sum approaches the original function $f(x) = x$.
2. Since it is a sum of $\sin$ terms, it is unavoidable that the sum equals $0$ at $x = \pm\pi$.
3. Most visible in the $N=100$ curve, sharp overshoots ("horns") appear near $x = \pm\pi$, jutting up and down.

In particular, item 3 — discovered in 1898 — is known as the Gibbs phenomenon, and its discovery led to significant later developments in mathematics; that story is for another time.

⑥ Summary

• A Fourier series represents a periodic function as a sum of trigonometric functions.
• The coefficients can be computed mechanically by definite integrals.
• It is a foundational technique in physics, engineering, and information science.