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---
title: "Schwartz distribution"
sort_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 such 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* $$\delta[\phi]$$
that map a test function $$\phi$$ from $$\mathcal{S}$$
to a number from $$\mathbb{C}$$;
this is often written as $$\Inprod{\delta}{\phi}$$.
This notation looks like the inner product of
a [Hilbert space](/know/concept/hilbert-space/),
and there is a good reason for that: 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 $$\Inprod{\delta}{\phi_n}$$
converges to $$\Inprod{\delta}{\phi}$$ for all $$\delta$$.
The power of the generalization is that $$f(x)$$
in fact 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 using partial integration:
$$\begin{aligned}
\Inprod{f'}{\phi}
= \int_{-\infty}^\infty f'(x) \: \phi(x) \dd{x}
= \Big[ f(x) \: \phi(x) \Big]_{-\infty}^\infty - \int_{-\infty}^\infty f(x) \: \phi'(x) \dd{x}
\end{aligned}$$
The test function removes the first term,
yielding the result $$- \Inprod{f}{\phi'}$$.
Although this was an example for a well-behaved $$f(x)$$,
we use it to define the derivative of any distribution $$\delta \in \mathcal{S}'$$:
$$\begin{aligned}
\boxed{
\Inprod{\delta'}{\phi}
\equiv - \Inprod{\delta}{\phi'}
}
\end{aligned}$$
Using the same trick, we can find the
[Fourier transform](/know/concept/fourier-transform/) (FT) of a distribution.
Let us define the FT like so,
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 well-behaved $$f(x)$$ is then as follows
(this is basically [Parseval's theorem](/know/concept/parsevals-theorem/)):
$$\begin{aligned}
\inprod{\tilde{f}}{\phi}
&= \int_{-\infty}^\infty \tilde{f}(k) \: \phi(k) \dd{k}
\\
&= \iint_{-\infty}^\infty f(x) \exp(- i k x) \: \phi(k) \dd{x} \dd{k}
\\
&= \int_{-\infty}^\infty f(x) \: \tilde{\phi}(x) \dd{x}
\\
&= \inprod{f}{\tilde{\phi}}
\end{aligned}$$
So we defined the FT and derived one of its properties,
but to generalize to Schwartz distributions $$\delta$$, we swap the roles:
the property is an axiom, and serves as the definition of the FT $$\tilde{\delta}$$:
$$\begin{aligned}
\boxed{
\inprod{\tilde{\delta}}{\phi}
\equiv \inprod{\delta}{\tilde{\phi}}
}
\end{aligned}$$
It can be shown that this generalization is a 1:1 mapping
between distributions in $$\mathcal{S}'$$.
## References
1. K.W. Jacobsen,
*Note on generalized functions (distributions)*, 2020,
unpublished.
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