From 7598e329f1b987e4d185741e82574b832b4ea319 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 27 Sep 2026 21:02:48 +0200
Subject: Improve knowledge base

---
 source/know/concept/schwartz-distribution/index.md | 58 ++++++++++++----------
 1 file changed, 32 insertions(+), 26 deletions(-)

(limited to 'source/know/concept/schwartz-distribution/index.md')

diff --git a/source/know/concept/schwartz-distribution/index.md b/source/know/concept/schwartz-distribution/index.md
index 2492d91..3d5186f 100644
--- a/source/know/concept/schwartz-distribution/index.md
+++ b/source/know/concept/schwartz-distribution/index.md
@@ -11,7 +11,7 @@ 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
+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.
@@ -31,13 +31,13 @@ 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 $$\mathcal{S}$$,
+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 $$\Inprod{f}{\phi}$$.
+this is often written as $$\Inprod{\delta}{\phi}$$.
 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
+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:
 
 $$\begin{aligned}
@@ -49,10 +49,11 @@ $$\begin{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 $$f$$.
+converges to $$\phi$$, then $$\Inprod{\delta}{\phi_n}$$
+converges to $$\Inprod{\delta}{\phi}$$ for all $$\delta$$.
 
-The power of this generalization is that $$f(x)$$ does not need to be well-behaved:
+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.
@@ -60,7 +61,7 @@ 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:
+we can work out what the derivative of a distribution is using partial integration:
 
 $$\begin{aligned}
     \Inprod{f'}{\phi}
@@ -68,20 +69,21 @@ $$\begin{aligned}
     = \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:
+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 $$\delta \in \mathcal{S}'$$:
 
 $$\begin{aligned}
     \boxed{
-        \Inprod{f'}{\phi} = - \Inprod{f}{\phi'}
+        \Inprod{\delta'}{\phi}
+        \equiv - \Inprod{\delta}{\phi'}
     }
 \end{aligned}$$
 
 Using the same trick, we can find the
-[Fourier transform](/know/concept/fourier-transform/) (FT)
-of a generalized function.
-We define the FT as follows,
+[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 $$x$$:
 
 $$\begin{aligned}
@@ -89,30 +91,34 @@ $$\begin{aligned}
     = \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:
+The FT of a well-behaved $$f(x)$$ is then as follows
+(this is basically [Parseval's theorem](/know/concept/parsevals-theorem/)):
 
 $$\begin{aligned}
     \inprod{\tilde{f}}{\phi}
     &= \int_{-\infty}^\infty \tilde{f}(k) \: \phi(k) \dd{k}
-    = \iint_{-\infty}^\infty f(x) \exp(- i k x) \: \phi(k) \dd{x} \dd{k}
+    \\
+    &= \iint_{-\infty}^\infty f(x) \exp(- i k x) \: \phi(k) \dd{x} \dd{k}
     \\
     &= \int_{-\infty}^\infty f(x) \: \tilde{\phi}(x) \dd{x}
-    = \inprod{f}{\tilde{\phi}}
+    \\
+    &= \inprod{f}{\tilde{\phi}}
 \end{aligned}$$
 
-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:
+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 $$\tilde{\delta}$$:
 
 $$\begin{aligned}
     \boxed{
-        \inprod{\tilde{f}}{\phi}
-        = \inprod{f}{\tilde{\phi}}
+        \inprod{\tilde{\delta}}{\phi}
+        \equiv \inprod{\delta}{\tilde{\phi}}
     }
 \end{aligned}$$
 
+It can be shown that this generalization is a 1:1 mapping
+between distributions in $$\mathcal{S}'$$.
+
 
 
 ## References
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