From c705ac1d7dc74709835a8c48fae4a7dd70dc5c49 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 25 Feb 2021 16:14:20 +0100
Subject: Expand knowledge base

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+---
+title: "Schwartz distribution"
+firstLetter: "S"
+publishDate: 2021-02-25
+categories:
+- Mathematics
+
+date: 2021-02-25T13:47:16+01:00
+draft: false
+markup: pandoc
+---
+
+# Schwartz distribution
+
+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 $\mathbb{C}$,
+which is often written as $\braket{f}{\phi}$.
+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:
+
+$$\begin{aligned}
+    f[\phi]
+    = \braket{f}{\phi}
+    = \int_{-\infty}^\infty f(x) \: \phi(x) \dd{x}
+\end{aligned}$$
+
+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 $\braket{f}{\phi_n}$
+converges to $\braket{f}{\phi}$ 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:
+
+$$\begin{aligned}
+    \braket{f'}{\phi}
+    = \int_{-\infty}^\infty f'(x) \: \phi(x) \dd{x}
+    = \Big[ f(x) \: \phi(x) \Big]_{-\infty}^\infty - \int_{-\infty}^\infty f(x) \: \phi'(x) \dd{x}
+\end{aligned}$$
+
+The test function removes the boundary term, yielding the result
+$- \braket{f}{\phi'}$. Although this was an example for a specific $f(x)$,
+we use it to define the derivative of any distribution:
+
+$$\begin{aligned}
+    \boxed{
+        \braket{f'}{\phi} = - \braket{f}{\phi'}
+    }
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \braket*{\tilde{f}}{\phi}
+    &= \int_{-\infty}^\infty \tilde{f}(k) \: \phi(k) \dd{k}
+    = \iint_{-\infty}^\infty f(x) \exp(- i k x) \: \phi(k) \dd{x} \dd{k}
+    \\
+    &= \int_{-\infty}^\infty f(x) \: \tilde{\phi}(x) \dd{x}
+    = \braket*{f}{\tilde{\phi}}
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \boxed{
+        \braket*{\tilde{f}}{\phi}
+        = \braket*{f}{\tilde{\phi}}
+    }
+\end{aligned}$$
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