---
title: "Schwartz distribution"
date: 2021-02-25
categories:
- Mathematics
layout: "concept"
---

A **Schwartz distribution**, also known as a **generalized function**,
is a generalization of a function,
allowing us to work with otherwise pathological definitions.

Notable examples of distributions are
the [Dirac delta function](/know/concept/dirac-delta-function/)
and the [Heaviside step function](/know/concept/heaviside-step-function/),
whose unusual properties are justified by this generalization.

We define the **Schwartz space** $\mathcal{S}$ of functions,
whose members are often called **test functions**.
Every such $\phi(x) \in \mathcal{S}$ must satisfy
the following constraint for any $p, q \in \mathbb{N}$:

$$\begin{aligned}
    \mathrm{max} \big| x^p \phi^{(q)}(x) \big| < \infty
\end{aligned}$$

In other words, a test function and its derivatives
decay faster than any polynomial.
Furthermore, all test functions must be infinitely differentiable.
These are quite strict requirements.

The **space of distributions** $\mathcal{S}'$ (note the prime)
is then said to consist of *functionals* $f[\phi]$
which map a test function $\phi$ from $\mathcal{S}$,
to a number from $\mathbb{C}$;
this is often written as $\Inprod{f}{\phi}$.
This notation looks like the inner product of
a [Hilbert space](/know/concept/hilbert-space/),
for good reason: any well-behaved function $f(x)$ can be embedded
into $\mathcal{S}'$ by defining the corresponding functional $f[\phi]$ as follows:

$$\begin{aligned}
    f[\phi]
    = \Inprod{f}{\phi}
    = \int_{-\infty}^\infty f(x) \: \phi(x) \dd{x}
\end{aligned}$$

Not all functionals qualify for $\mathcal{S}'$:
they also need to be linear in $\phi$, and **continuous**,
which in this context means: if a series $\phi_n$
converges to $\phi$, then $\Inprod{f}{\phi_n}$
converges to $\Inprod{f}{\phi}$ for all $f$.

The power of this generalization is that $f(x)$ does not need to be well-behaved:
for example, the Dirac delta function can also be used,
whose definition is nonsensical *outside* of an integral,
but perfectly reasonable *inside* one.
By treating it as a distribution,
we gain the ability to sanely define e.g. its derivatives.

Using the example of embedding a well-behaved function $f(x)$ into $\mathcal{S}$,
we can work out what the derivative of a distribution is:

$$\begin{aligned}
    \Inprod{f'}{\phi}
    = \int_{-\infty}^\infty f'(x) \: \phi(x) \dd{x}
    = \Big[ f(x) \: \phi(x) \Big]_{-\infty}^\infty - \int_{-\infty}^\infty f(x) \: \phi'(x) \dd{x}
\end{aligned}$$

The test function removes the boundary term, yielding the result
$- \Inprod{f}{\phi'}$. Although this was an example for a specific $f(x)$,
we use it to define the derivative of any distribution:

$$\begin{aligned}
    \boxed{
        \Inprod{f'}{\phi} = - \Inprod{f}{\phi'}
    }
\end{aligned}$$

Using the same trick, we can find the
[Fourier transform](/know/concept/fourier-transform/) (FT)
of a generalized function.
We define the FT as follows,
but be prepared for some switching of the names $k$ and $x$:

$$\begin{aligned}
    \tilde{\phi}(x)
    = \int_{-\infty}^\infty \phi(k) \exp(- i k x) \dd{k}
\end{aligned}$$

The FT of a Schwartz distribution $f$ then turns out to be as follows:

$$\begin{aligned}
    \inprod{\tilde{f}}{\phi}
    &= \int_{-\infty}^\infty \tilde{f}(k) \: \phi(k) \dd{k}
    = \iint_{-\infty}^\infty f(x) \exp(- i k x) \: \phi(k) \dd{x} \dd{k}
    \\
    &= \int_{-\infty}^\infty f(x) \: \tilde{\phi}(x) \dd{x}
    = \inprod{f}{\tilde{\phi}}
\end{aligned}$$

Note that the ordinary FT $\tilde{f}(k) = \hat{\mathcal{F}}\{f(x)\}$ is
already a 1:1 mapping of test functions $\phi \leftrightarrow \tilde{\phi}$.
As it turns out,
in this generalization it is also a 1:1 mapping of distributions in $\mathcal{S}'$,
defined as:

$$\begin{aligned}
    \boxed{
        \inprod{\tilde{f}}{\phi}
        = \inprod{f}{\tilde{\phi}}
    }
\end{aligned}$$



## References
1.  K.W. Jacobsen,
    *Note on generalized functions (distributions)*, 2020,
    unpublished.