Theory of distributions

Jan Riopedre, Arnau Rojals and F. Javier Rodríguez

Example

Fourier Transform

Fourier Transform

720 x 1280 = 921600 coefficients

Fourier Transform

n = 2

n = 4

n = 8

n = 16

n = 32

n = 64

n = 128

n = 256

n = 512

n = 1024

n = 2048

n = 4096

n = 8192

n = 16384

n = 32768

Test Functions

Test Functions

Motivated by $\delta(x)$ approach as multiplyng a ‘suitably smooth’ function.

Test Functions

Motivated by $\delta(x)$ approach as multiplyng a ‘suitably smooth’ function.

Test Functions

We say that $\phi(x)$ is a test function if:

  • $\phi(x)$ is a $\mathcal{C}^{\infty}$ function
  • $\phi(x)$ has compact support.
Condition 1 implies that all the derivatives of a test function are also test functions
As a note, test functions do exist.

Test Functions

Example: $$\Phi(x) = \begin{cases}0 \hspace{40.5mm}x\leq 0\\e^{-1/x} \hspace{20mm}x > 0\end{cases}$$
It is $\mathcal{C}^\infty$, but it does not have compact support


$$\Rightarrow \phi(x) = \Phi(x)\Phi(1-x)$$

Test Functions

Convergence for $\{\phi_n(x)\}$:
We say that $\phi_n(x) \to 0$ as $n \to \infty$ if

  • $\phi_n(x)$ and all its derivatives $\phi_n^{(m)}(x)$ tend to zero, uniformly in both $x$ and $m$

  • There is an interval $(a, b)$ containing the support of all the $\phi_n$.

The action of a test function

Action of $f$ on a test function $\phi(x)$

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

Properties

  • $\langle f, a\phi+b\psi\rangle = a\langle f, \phi\rangle + b \langle f, \psi\rangle$
  • $$\text{if } \phi_n(x) \to 0 \text{ then } \langle f, \phi_n \rangle \to 0$$

Distribution

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

Why?
 
Take the of function further
 
More broader and general

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

 

A distribution $( D )$ is a continuous linear map from the space of test functions to $( \mathbb{R} )$

$\mathcal{D} : \phi \mapsto \langle \mathcal{D}, \phi \rangle \in \mathbb{R}$

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

  • $ \langle \mathcal{D}, a \varphi + b \psi \rangle = a \langle \mathcal{D}, \varphi \rangle + b \langle \mathcal{D}, \psi \rangle $

  • if $ \varphi_n(x) \to 0 \quad \text{as} \quad n \to \infty, $
    then $ \langle \mathcal{D}, \varphi_n \rangle \to 0 $

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

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

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

$ \langle \delta, \phi \rangle = \phi(0) $
$= \int_{-\infty}^{\infty} \delta(x) \phi(x) \, dx $

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

 
  • $ \langle \mathcal{D} + \mathcal{E}, \phi \rangle = \langle \mathcal{D}, \phi \rangle + \langle \mathcal{E}, \phi \rangle $
  • $ \langle a \mathcal{D}, \phi \rangle = a \langle \mathcal{D}, \phi \rangle $
  • $ \langle \mathcal{D}(x - a), \phi(x) \rangle = \langle \mathcal{D}(x), \phi(x + a) \rangle $
  • $ \langle \mathcal{D}(ax), \phi(x) \rangle = \frac{1}{|a|} \langle \mathcal{D}, \phi(x/a) \rangle $
  • $ \langle \Phi(x) \mathcal{D}(x), \phi(x) \rangle = \langle \mathcal{D}(x), \Phi(x) \phi(x) \rangle $

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

  • $\langle \mathcal{D}^‘, \phi \rangle = -\langle \mathcal{D}, \phi’ \rangle$

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

  • $\langle \mathcal{D}^‘, \phi \rangle = -\langle \mathcal{D}, \phi’ \rangle$

  • Example: $\mathcal{H}'(x) = \delta(x)$

Distribution

Motivation
Definition
Linearity and continuity
The delta function
Properties
The derivative of a distribution

  • $\langle \mathcal{D}^‘, \phi \rangle = -\langle \mathcal{D}, \phi’ \rangle$

  • Example: $\mathcal{H}'(x) = \delta(x)$

  • $\langle \mathcal{D}^{(m)}(x), \phi(x) \rangle = (-1)^m \langle \mathcal{D}, \phi^{(m)}(x) \rangle$

Extensions of the theory of distributions

More variables

 

  • Multivariable test function
$$\mathcal{C}^{\infty} \text{ and with compact support in all the arguments}$$

More variables

  • Multivariable test function

$$\mathcal{C}^{\infty} \text{ and with compact support in all the arguments}$$

  • Multivariable distribution
Continuous linear map from the space of multivariable test functions to real numbers
Example: $\langle \delta(\mathbf{x}), \phi(\mathbf{x}) \rangle = \phi(0)$

More variables

  • Derivatives
$$ \left\langle \frac{\partial \mathcal{D}}{\partial x_i}, \, \phi \right\rangle = - \left\langle \mathcal{D}, \, \frac{\partial \phi}{\partial x_i} \right\rangle $$
$$ \nabla \wedge \nabla \mathcal{D} \equiv 0 $$

Fourier transforms

  • Concept and deduction

$$\hat{\phi}(k) = \int_{-\infty}^{\infty} \phi(x) e^{ikx} , dx$$

Fourier transforms

  • Tempered distributions
Test functions before: with compact support and $\mathcal{C}^\infty$

Fourier transforms

  • Tempered distributions
Test functions before: with compact support and $\mathcal{C}^\infty$
↓
all the derivatives decay faster than any power of x

Fourier transforms

  • Tempered distributions

$$\phi(x) \text{ test function} \quad \Rightarrow \quad \hat{\phi}(k) \text{ test function}$$

$\hat{\phi}(k) = \int_{-\infty}^{\infty} \phi(x) e^{ikx} dx$

$\hat{\psi}(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \psi(k) e^{-ikx} dk$

$\frac{d\hat{\phi}}{dx} = -ik\hat{\phi}$

$\hat{x\phi} = -i \frac{d\hat{\phi}}{dk}$

Fourier transforms

  • Tempered distributions

The action of the Fourier transform of an ordinary function on a test function:

$$\langle \hat{f}, \phi \rangle = \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} f(x) e^{ikx} dx \right) \phi(k) dk$$

$$ \hspace{27mm}= \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} \phi(k) e^{ikx} dk \right) f(x) dx $$

$$ \hspace{-55mm}= \langle f, \hat{\phi} \rangle. $$

So we define: $\hspace{5mm} \langle \hat{\mathcal{D}}, \phi \rangle = \langle \mathcal{D}, \hat{\phi} \rangle$

$\hspace{68mm}\langle \check{\mathcal{D}}, \phi \rangle = \langle \mathcal{D}, \check{\phi} \rangle$

Fourier transforms

  • Tempered distributions

The Fourier transform of the derivative $\mathcal{D}’ = \frac{d\mathcal{D}}{dx}$ is $-ik\hat{\mathcal{D}}$

$ \langle \hat{\mathcal{D}^‘}, \phi \rangle = \langle \mathcal{D}’, \hat{\phi} \rangle $ $ = -\langle \mathcal{D}, \frac{d \hat{\phi}}{dk} \rangle $ $ \hspace{28mm}= -\langle \mathcal{D}, ix \hat{\phi} \rangle \ = \langle -ik \hat{\mathcal{D}}, \phi \rangle $

So we define: $\hspace{5mm} \langle \hat{\mathcal{D}}, \phi \rangle = \langle \mathcal{D}, \hat{\phi} \rangle$

$\hspace{68mm}\langle \check{\mathcal{D}}, \phi \rangle = \langle \mathcal{D}, \check{\phi} \rangle$

Fourier transforms

  • Two Examples

Fourier transforms

  • FT of delta
Informally,

$$ \int_{-\infty}^{\infty} \delta(x) e^{ikx} dx = e^{ik \cdot 0} = 1 $$

Formally,

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

$$ \hat{\delta}(k) = 1 $$

Fourier transforms

  • FT of 1
For the inverse, we have:

$$ \check{\delta} = \frac{1}{2\pi} \int_{-\infty}^{\infty} \delta(k) e^{-ikx} , dk \ = \frac{1}{2\pi}, $$

Formally,

$$ \overset{({\check{\delta}}\hat)=\delta}{\Rightarrow} \quad \hat{1}(k) = 2\pi \delta(k) $$

Example: Heat Equation

Consider:

$$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad -\infty < x < \infty, \quad t > 0, $$

$$ u(x, 0) = \delta(x). $$

Taking the Fourier transform in $x$. The equation for $ \hat{u}(k, t)$ is:

$$ \frac{\partial \hat{u}}{\partial t} = -k^2 \hat{u}, \quad -\infty < k < \infty, \quad t > 0, $$

$$ u(x, 0) = \delta(k) = 1. $$

Example: Heat Equation

Taking the Fourier transform in $x$. The equation for $ \hat{u}(k, t)$ is:

$$ \frac{\partial \hat{u}}{\partial t} = -k^2 \hat{u}, \quad -\infty < k < \infty, \quad t > 0, $$

$$ u(x, 0) = \delta(k) = 1. $$

The solution is:

$$ \hat{u}(k, t) = exp(-k^2t) \Rightarrow u(x, t) = \frac{1}{2\sqrt{\pi t}} exp(-x^2 / 4t). $$

Heat Equation Solution

Questions?