Chapter 4 Fourier Series
Fourier series arise naturally when constructing solutions of certain initial-boundary value problems of partial differential equations (PDEs). For example, they arise from studying the initial-boundary value problem for the heat equation
\begin{equation*}
\left\{
\begin{aligned}
u_t(x,t)-u_{xx}(x,t) \amp = 0 \amp 0\lt x \lt l, t>0\\
u(0,t)\amp =u(l,t)=0 \amp t>0\\
u(x, 0)\amp =g(x) \amp 0\lt x \lt l
\end{aligned}
\right.
\end{equation*}
where the initial data \(g(x)\) is a given continuous function on \([0, l]\text{,}\) and traditionally we would like the solution \(u(x, t)\) to be twice continuously differentiable in \(x\text{,}\) once continuously differentiable in \(t\) in the domain \((0, l)\times (0,\infty)\text{,}\) and continuous on \([0, l]\times [0,\infty)\text{.}\)
There is an elementary procedure of looking for separable particular solutions \(u(x, t)\) of the form \(X(x)T(t)\text{,}\) which solves the homogeneous heat equation and the homogeneous boundary conditions. The result is that for any \(n\in \mathbb N\text{,}\)
\begin{equation*}
u_{n}(x, t) := \sin \left(\frac{n \pi x}{l}\right) e^{- \left(\frac{n \pi}{l}\right)^2 t}
\end{equation*}
is such a solution. Since we are so far dealing with linear homogeneous equations, any linear combination of solutions is still a solution, so
\begin{equation*}
\sum_{n \in \text{a finite set}} c_n \sin \left(\frac{n \pi x}{l}\right) e^{- \left(\frac{n \pi}{l}\right)^2 t}
\end{equation*}
also satisfies the same equations. What remains is whether one can choose the \(c_{n}\)’s so that this solution at \(t=0\) gives rise to the prescribed initial data \(g(x)\text{.}\)
For that purpose, first we need to form an infinite sum and demand that
\begin{equation}
\sum_{n=1}^{\infty} c_n \sin \left(\frac{n \pi x}{l} \right)= g(x) \quad \text{on } (0,l)
\text{ in an appropriate sense}.\tag{4.0.1}
\end{equation}
But we also need to make sense of the infinite series as a continuously differentiable solution.
The expansion (4.0.1) is a version of the Fourier series expansion. It turns out that we must choose \(c_{n}\) such that
\begin{equation}
c_{n}= \frac{2}{l} \int_{0}^l g(x) \sin \left(\frac{n \pi x}{l}\right) \, dx.\tag{4.0.2}
\end{equation}
The key properties that lead to the formula (4.0.2) for \(c_{n}\) and many other properties of the series are the orthogonality relations of the terms over the interval \([0, l]\) stated below:
\begin{equation*}
\int_{0}^{l} \sin \left(\frac{n \pi x}{l}\right) \sin \left(\frac{m \pi x}{l}\right) \, dx =
\begin{cases} 0 \amp \text{if $n\ne m$,} \\
\frac l2 \amp \text{if $n= m$.}
\end{cases}
\end{equation*}
If the most elementary notion of pointwise convergence is used in (4.0.1), then it is not so easy to justify the interchange of summation and integration. It turns out that a more useful notion of convergence in this context is that of mean square convergence defined as
\begin{equation*}
\lim_{N\to \infty} \int_{0}^{l}\vert g(x) - \sum_{n=1}^{N} c_n \sin \left(\frac{n \pi x}{l} \right)\vert^{2} \, dx
\to 0.
\end{equation*}
