Fourier Transform of Distributions | Fourier Analysis (Advanced)

Q: What is a distribution (generalized function)?

A distribution is a generalized "function" defined as a continuous linear functional that assigns a number to each test function (a smooth function with compact support). It lets us handle objects that are not functions in the ordinary sense, such as the Dirac delta $\delta(x)$ (evaluation at the point $x=0$).

Q: What properties does the Dirac delta have?

The delta $\delta$ is defined on a test function $\phi$ by $\langle \delta, \phi\rangle = \phi(0)$. Formally, $\int_{-\infty}^{\infty}\delta(x)\phi(x)\,dx = \phi(0)$. Its Fourier transform is the constant function $\hat{\delta}(\xi)=1$. In physics it models point charges, point impulses, and the like.

Q: How is distribution theory useful in Fourier analysis?

Within the framework of tempered distributions, every tempered distribution has a Fourier transform (with no restriction to $L^1$ or $L^2$). This lets us define the Fourier transforms of the $\delta$ function, constant functions, polynomials, and so on, providing the mathematical foundation for PDE analysis (Green's functions), signal processing (impulse response), and quantum mechanics (position and momentum operators).

Fourier Transform of Distributions

Difficulty: Advanced

Extending Fourier analysis through distribution theory

Introduction

Some "functions", such as the delta function and constant functions, have no Fourier transform in the classical sense. The theory of distributions (generalized functions) provides a framework in which the Fourier transform can be defined rigorously even for these objects.

Test Functions

A distribution is defined by its action on test functions. General distributions act on compactly supported smooth functions $C_c^\infty(\mathbb{R})$, but to keep the Fourier transform closed within the space, this chapter uses the larger Schwartz space $\mathcal{S}(\mathbb{R})$ as the test functions and works with its dual, the tempered distributions.

The Schwartz space $\mathcal{S}(\mathbb{R})$

The space of rapidly decreasing functions:

$$\mathcal{S}(\mathbb{R}) = \left\{\varphi \in C^\infty(\mathbb{R}) \,\middle|\, \sup_x |x^\alpha \varphi^{(\beta)}(x)| < \infty \text{ for all } \alpha, \beta \geq 0\right\}$$

A function belongs to the Schwartz space if it is:

infinitely differentiable;
such that every derivative decays faster than any inverse power of $x$ (i.e. $x^\alpha \varphi^{(\beta)}(x)$ stays bounded for all $\alpha, \beta$).

Examples

$\varphi(x) = e^{-x^2}$: belongs to the Schwartz space.
$\varphi(x) = e^{-|x|}$: does not belong to the Schwartz space (it has a point at the origin where it is not differentiable).
$\varphi(x) = \dfrac{1}{1+x^2}$: infinitely differentiable, but its decay is too slow (like $x^{-2}$), so it does not belong to the Schwartz space — smoothness alone does not imply membership in $\mathcal{S}$.

Fig. 1. Examples of test functions. Since $e^{-x^2}$ (blue) is infinitely differentiable and rapidly decreasing, it belongs to the Schwartz space $\mathcal{S}(\mathbb{R})$. By contrast, $e^{-|x|}$ (red) has a corner at the origin and is not differentiable there, so it does not belong to $\mathcal{S}(\mathbb{R})$.

Tempered Distributions

The space of tempered distributions $\mathcal{S}'(\mathbb{R})$

The space of all continuous linear functionals on $\mathcal{S}(\mathbb{R})$.

A distribution $T \in \mathcal{S}'$ returns a value $\langle T, \varphi \rangle \in \mathbb{C}$ for each test function $\varphi \in \mathcal{S}$.

Relation to ordinary functions

If $f$ is a function of at most polynomial growth, then

$$\langle T_f, \varphi \rangle = \displaystyle\int_{-\infty}^{\infty}f(x)\varphi(x)\,dx$$

defines a tempered distribution $T_f \in \mathcal{S}'$.

A Rigorous Definition of the Delta Function

The delta distribution

$$\langle \delta, \varphi \rangle = \varphi(0)$$

The intuitive picture that $\delta(x)$ is “infinite” at $x = 0$ and $0$ elsewhere is made rigorous through the action of the distribution.

$\delta(x)$ $\displaystyle\int_{-\infty}^{\infty}\!\delta(x)\,dx = 1$ $= 0$

Fig. 2. Intuitive picture of the delta function: "infinite" at $x=0$, $0$ elsewhere, with total integral $1$. The arrow represents the "weight" of the impulse (i.e. the integral value $1$). Rigorously, it is defined by its action on test functions, $\langle\delta,\varphi\rangle=\varphi(0)$.

Properties of the delta function

$\langle \delta_a, \varphi \rangle = \varphi(a)$ (shifted delta function)
$\langle \delta', \varphi \rangle = -\varphi'(0)$ (derivative of the delta function)
$x\delta(x) = 0$ (in the sense of distributions)

The Fourier Transform of Distributions

Throughout, the Fourier transform of an ordinary function is taken to be $\hat{f}(\omega) = \displaystyle\int_{-\infty}^{\infty} f(x)\,e^{-i\omega x}\,dx$ (the angular-frequency convention, with no constant on the forward transform). Under this convention a factor $\dfrac{1}{2\pi}$ appears in the inverse transform, and a $2\pi$ shows up in results such as $\hat{1} = 2\pi\delta$.

The Fourier transform $\mathcal{F}: \mathcal{S} \to \mathcal{S}$ on the Schwartz space is a bijection.

Fourier transform of a distribution

For $T \in \mathcal{S}'$, we define $\hat{T} \in \mathcal{S}'$ by

$$\langle \hat{T}, \varphi \rangle = \langle T, \hat{\varphi} \rangle$$

for every $\varphi \in \mathcal{S}$.

Example: Fourier transform of the delta function

$$\langle \hat{\delta}, \varphi \rangle = \langle \delta, \hat{\varphi} \rangle = \hat{\varphi}(0) = \displaystyle\int_{-\infty}^{\infty}\varphi(x)\,dx = \langle 1, \varphi \rangle$$

Therefore $\hat{\delta} = 1$ (the constant function $1$).

Example: Fourier transform of a constant

$\hat{1} = 2\pi\delta$

Operations on Distributions

Differentiation

The derivative of a distribution is always defined:

$$\langle T', \varphi \rangle = -\langle T, \varphi' \rangle$$

Convolution

Under suitable conditions, the convolution of distributions can also be defined:

$$(T * \varphi)(x) = \langle T, \varphi(x - \cdot) \rangle$$

In particular, $\delta * f = f$ (the delta function is the identity for convolution).

Properties of the Fourier transform

The same properties continue to hold in the world of distributions:

$\mathcal{F}[T'] = i\omega \hat{T}$
$\mathcal{F}[xT] = i\dfrac{d\hat{T}}{d\omega}$
$\mathcal{F}[T * S] = \hat{T} \cdot \hat{S}$ (under suitable conditions)

Important Examples

$T$	$\hat{T}$
$\delta(x)$	$1$
$1$	$2\pi\delta(\omega)$
$e^{i\omega_0 x}$	$2\pi\delta(\omega - \omega_0)$
$\text{sgn}(x)$	$\operatorname{p.v.}\dfrac{2}{i\omega}$ (principal value)
$H(x)$ (Heaviside)	$\pi\delta(\omega) + \operatorname{p.v.}\dfrac{1}{i\omega}$
$\displaystyle\sum_n \delta(x - nT)$	$\dfrac{2\pi}{T}\displaystyle\sum_k \delta(\omega - \dfrac{2\pi k}{T})$

Illustrations of the transform pairs

The transform pairs above can be illustrated as follows (left: time domain $T$; right: frequency domain $\hat{T}$). A pink upward arrow ↑ represents a delta function (an impulse), while blue curves and lines represent ordinary functions.

$\delta(x)$ $\mathcal{F}$ $1$ $x$ $\omega$

Fig. 3. $\delta(x)\ \xrightarrow{\ \mathcal{F}\ }\ 1$. A single-point impulse in the time domain is transformed into a constant spectrum that contains all frequencies equally.

$1$ $\mathcal{F}$ $2\pi\delta(\omega)$ $x$ $\omega$

Fig. 4. $1\ \xrightarrow{\ \mathcal{F}\ }\ 2\pi\delta(\omega)$. A constant function, which carries only a DC component (frequency $0$), maps to a single-point spectrum standing at the origin (the dual of Fig. 3).

$e^{i\omega_0 x}$ $\mathcal{F}$ $2\pi\delta(\omega-\omega_0)$ $\omega_0$ $x$ $\omega$

Fig. 5. $e^{i\omega_0 x}\ \xrightarrow{\ \mathcal{F}\ }\ 2\pi\delta(\omega-\omega_0)$. A complex sinusoid of a single frequency $\omega_0$ becomes one line spectrum standing at $\omega_0$.

$\operatorname{sgn}(x)$ $+1$ $-1$ $\mathcal{F}$ $\dfrac{2}{i\omega}$ $x$ $\omega$

Fig. 6. $\operatorname{sgn}(x)\ \xrightarrow{\ \mathcal{F}\ }\ \dfrac{2}{i\omega}$ (principal value). The sign function, being odd, has a purely imaginary odd spectrum that diverges as $\omega\to 0$ (interpreted in the sense of the Cauchy principal value).

$H(x)$ $1$ $\mathcal{F}$ $\pi\delta(\omega)$ $\dfrac{1}{i\omega}$ $x$ $\omega$

Fig. 7. $H(x)\ \xrightarrow{\ \mathcal{F}\ }\ \pi\delta(\omega)+\dfrac{1}{i\omega}$. The Heaviside step function becomes the sum of a DC component $\pi\delta(\omega)$ and an odd, purely imaginary component $1/(i\omega)$ (corresponding to $H=\tfrac12(1+\operatorname{sgn})$).

$T$ $\mathcal{F}$ $\dfrac{2\pi}{T}$ $x$ $\omega$

Fig. 8. $\displaystyle\sum_n \delta(x-nT)\ \xrightarrow{\ \mathcal{F}\ }\ \dfrac{2\pi}{T}\sum_k \delta\!\left(\omega-\dfrac{2\pi k}{T}\right)$. A train of equally spaced impulses with spacing $T$ (a Dirac comb) is transformed into a train of impulses with spacing $2\pi/T$. The narrower the spacing in time, the wider the spacing in frequency (the Poisson summation formula).

On "the sense of the principal value"

The $\dfrac{1}{i\omega}$ appearing in the transforms of $\operatorname{sgn}(x)$ and $H(x)$ diverges at $\omega=0$, and $1/\omega$ cannot be integrated in the usual way near the origin. We therefore interpret it as the principal-value distribution $\operatorname{p.v.}\dfrac{1}{\omega}$, obtained by removing a symmetric interval around the origin and then taking the limit:

$$\left\langle \operatorname{p.v.}\tfrac{1}{\omega},\ \varphi\right\rangle=\lim_{\varepsilon\to 0^+}\int_{|\omega|\ge \varepsilon}\frac{\varphi(\omega)}{\omega}\,d\omega.$$

Since $1/\omega$ is an odd function, the divergences on the left and right cancel symmetrically and yield a finite value. This is the Cauchy principal value.

Summary

Schwartz space $\mathcal{S}$: rapidly decreasing, infinitely differentiable functions.
Tempered distributions $\mathcal{S}'$: continuous linear functionals on $\mathcal{S}$.
Delta function: $\langle \delta, \varphi \rangle = \varphi(0)$.
Fourier transform of a distribution: $\langle \hat{T}, \varphi \rangle = \langle T, \hat{\varphi} \rangle$.
$\hat{\delta} = 1$, $\hat{1} = 2\pi\delta$.

Frequently Asked Questions

What is a distribution (generalized function)?

A distribution is a generalized "function" defined as a continuous linear functional that assigns a number to each test function (a smooth function with compact support). It lets us handle objects that are not functions in the ordinary sense, such as the Dirac delta $\delta(x)$ (evaluation at the point $x=0$).

What properties does the Dirac delta have?

The delta $\delta$ is defined on a test function $\phi$ by $\langle \delta, \phi\rangle = \phi(0)$. Formally, $\int_{-\infty}^{\infty}\delta(x)\phi(x)\,dx = \phi(0)$. Its Fourier transform is the constant function $\hat{\delta}(\xi)=1$. In physics it models point charges, point impulses, and the like.

How is distribution theory useful in Fourier analysis?

Within the framework of tempered distributions, every tempered distribution has a Fourier transform (with no restriction to $L^1$ or $L^2$). This lets us define the Fourier transforms of the $\delta$ function, constant functions, polynomials, and so on, providing the mathematical foundation for PDE analysis (Green's functions), signal processing (impulse response), and quantum mechanics (position and momentum operators).