Fourier Analysis - Core Definitions

Fourier analysis decomposes functions into sinusoidal components, providing a powerful framework for understanding periodic phenomena, wave propagation, and signal processing in physics.

Fourier Series

DefinitionFourier Series

A periodic function $f(x)$ with period $2L$ can be expanded as:

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

where the Fourier coefficients are:

$a_n = \frac{1}{L}\int_{-L}^L f(x)\cos\left(\frac{n\pi x}{L}\right)dx$

$b_n = \frac{1}{L}\int_{-L}^L f(x)\sin\left(\frac{n\pi x}{L}\right)dx$

These formulas follow from the orthogonality relations of sine and cosine functions over the interval $[-L, L]$ .

ExampleSquare Wave

For the square wave: $f(x) = \begin{cases}1 & 0 < x < \pi \\ -1 & -\pi < x < 0\end{cases}$

with period $2\pi$ , by symmetry $a_n = 0$ for all $n$ . The sine coefficients are:

$b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin(nx)dx = \frac{2}{\pi}\int_0^{\pi}\sin(nx)dx = \begin{cases}\frac{4}{n\pi} & n \text{ odd} \\ 0 & n \text{ even}\end{cases}$

Thus: $f(x) = \frac{4}{\pi}\left(\sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \cdots\right)$

The Gibbs phenomenon appears at discontinuities: the series overshoots by about 9% regardless of how many terms are included.

DefinitionComplex Fourier Series

Using Euler's formula $e^{inx} = \cos(nx) + i\sin(nx)$ , the Fourier series can be written compactly as:

$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{in\pi x/L}$

where:

$c_n = \frac{1}{2L}\int_{-L}^L f(x)e^{-in\pi x/L}dx$

This form makes the symmetry between positive and negative frequencies explicit and simplifies many calculations.

Fourier Transform

DefinitionFourier Transform

For a function $f(x)$ defined on $\mathbb{R}$ (not necessarily periodic), the Fourier transform is:

$\tilde{f}(k) = \mathcal{F}[f](k) = \int_{-\infty}^{\infty} f(x)e^{-ikx}dx$

The inverse Fourier transform is:

$f(x) = \mathcal{F}^{-1}[\tilde{f}](x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \tilde{f}(k)e^{ikx}dk$

Alternative conventions exist with different factors of $2\pi$ distributed between forward and inverse transforms.

ExampleGaussian Function

The Gaussian $f(x) = e^{-x^2/(2\sigma^2)}$ has Fourier transform:

$\tilde{f}(k) = \sqrt{2\pi}\sigma e^{-\sigma^2 k^2/2}$

This shows that Gaussians are eigenfunctions of the Fourier transform (up to scaling). Moreover, narrow Gaussians in $x$ -space correspond to wide Gaussians in $k$ -space, illustrating the uncertainty principle:

$\Delta x \cdot \Delta k \geq \frac{1}{2}$

In quantum mechanics with $k = p/\hbar$ , this becomes Heisenberg's uncertainty principle.

Properties of Fourier Transforms

TheoremFundamental Properties

The Fourier transform has several key properties:

Linearity: $\mathcal{F}[af + bg] = a\mathcal{F}[f] + b\mathcal{F}[g]$

Translation: $\mathcal{F}[f(x - a)](k) = e^{-ika}\tilde{f}(k)$

Modulation: $\mathcal{F}[e^{ik_0 x}f(x)](k) = \tilde{f}(k - k_0)$

Scaling: $\mathcal{F}[f(ax)](k) = \frac{1}{|a|}\tilde{f}(k/a)$

Differentiation: $\mathcal{F}[f'(x)](k) = ik\tilde{f}(k)$

Convolution: $\mathcal{F}[f * g](k) = \tilde{f}(k)\tilde{g}(k)$

The differentiation property transforms differential equations into algebraic equations, while the convolution theorem is central to linear response theory and Green's function methods.

ExampleHeat Equation Solution

The heat equation $\partial_t u = \alpha\nabla^2 u$ with initial condition $u(x, 0) = f(x)$ transforms to:

$\frac{\partial\tilde{u}}{\partial t} = -\alpha k^2\tilde{u}(k,t)$

This gives $\tilde{u}(k,t) = \tilde{f}(k)e^{-\alpha k^2 t}$ . Inverting:

$u(x,t) = \frac{1}{\sqrt{4\pi\alpha t}}\int_{-\infty}^{\infty} f(x')e^{-(x-x')^2/(4\alpha t)}dx'$

This is the convolution of the initial condition with the heat kernel $G(x,t) = e^{-x^2/(4\alpha t)}/\sqrt{4\pi\alpha t}$ .

RemarkParseval's Theorem

The total energy is preserved under Fourier transform:

$\int_{-\infty}^{\infty}|f(x)|^2dx = \frac{1}{2\pi}\int_{-\infty}^{\infty}|\tilde{f}(k)|^2dk$

In quantum mechanics, this ensures that probability is conserved in both position and momentum representations.

Fourier analysis transforms problems in physical space to momentum/frequency space, often simplifying differential equations and revealing hidden symmetries in physical systems.