Fourier Transform Methods

The Fourier transform of a function \(f\) is denoted either by \(\mathcal{F}\{f(x)\}\) or \(\bar{f}(\xi)\) and is defined as

\[ \bar{f}(\xi)=\int\limits_{-\infty}^\infty f(x)e^{i\xi x}\mathrm{d}x. \]

The Fourier inversion theorem states that provided the inverse exists, it is given by

\[ f(x)=\frac{1}{2\pi}\int\limits_{-\infty}^\infty\bar{f}(\xi)e^{-i\xi x}\mathrm{d}\xi. \]

In the lectures we have discussed different functional spaces and how this impacts notions like existence (i.e. does the integral converge?)

Important results

In the lectures we have met a number of important results that are useful when solving Fourier transform problems. These include the derivative theorem and the convolution theorem.

Derivative Theorem

The derivative theorem states that provided \(f\in L^1(\mathbb{R})\),

\[ \mathcal{F}\{f'(x)\}=-i\xi\mathcal{F}\{f(x)\}=-i\xi\bar{f}(\xi). \]

Convolution Theorem

The convolution theorem states that

\[ \mathcal{F}\{f*g\}=\mathcal{F}\{f\}\mathcal{F}\{g\}, \]

where \(f*g\) denotes the convolution of \(f\) and \(g\), which is defined by

\[\begin{split} \begin{aligned} (f*g)(x)&=\int\limits_{-\infty}^\infty f(y)g(x-y)\mathrm{d}y \\ &=\int\limits_{-\infty}^\infty f(x-y)g(y)\mathrm{d}y. \end{aligned} \end{split}\]

The heat equation Cauchy problem on an infinitely long rod

The heat equation with initial conition \(u(x,0)=f(x)\) has solution given by

\[ u(x,t)=\frac{1}{2c\sqrt{\pi t}}\int\limits_{-\infty}^\infty {e^{-\frac{(x-y)^2}{4c^2t}}f(y)\mathrm{d}y}. \]

You don’t need to remember this formula for the exam, but you should know how to derive it by using Fourier transforms; the video below revises the derivation we saw in the lectures.

Heat equation: solution using the Fourier transform