Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 23 insertions, 23 deletions

diff --git a/source/know/concept/schwartz-distribution/index.md b/source/know/concept/schwartz-distribution/index.md
index e74dc4a..2492d91 100644
--- a/source/know/concept/schwartz-distribution/index.md
+++ b/source/know/concept/schwartz-distribution/index.md
@@ -16,10 +16,10 @@ 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,
+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 $p, q \in \mathbb{N}$:
+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
@@ -30,15 +30,15 @@ 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 $\Inprod{f}{\phi}$.
+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:
+for good reason: any well-behaved function $$f(x)$$ can be embedded
+into $$\mathcal{S}'$$ by defining the corresponding functional $$f[\phi]$$ as follows:
 
 $$\begin{aligned}
     f[\phi]
@@ -46,20 +46,20 @@ $$\begin{aligned}
     = \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 $f$.
+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:
+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 $\mathcal{S}$,
+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:
 
 $$\begin{aligned}
@@ -69,7 +69,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The test function removes the boundary term, yielding the result
-$- \Inprod{f}{\phi'}$. Although this was an example for a specific $f(x)$,
+$$- \Inprod{f}{\phi'}$$. Although this was an example for a specific $$f(x)$$,
 we use it to define the derivative of any distribution:
 
 $$\begin{aligned}
@@ -82,14 +82,14 @@ Using the same trick, we can find the
 [Fourier transform](/know/concept/fourier-transform/) (FT)
 of a generalized function.
 We define the FT as follows,
-but be prepared for some switching of the names $k$ and $x$:
+but be prepared for some switching of the names $$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:
+The FT of a Schwartz distribution $$f$$ then turns out to be as follows:
 
 $$\begin{aligned}
     \inprod{\tilde{f}}{\phi}
@@ -100,10 +100,10 @@ $$\begin{aligned}
     = \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 $\phi \leftrightarrow \tilde{\phi}$.
+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 $\mathcal{S}'$,
+in this generalization it is also a 1:1 mapping of distributions in $$\mathcal{S}'$$,
 defined as:
 
 $$\begin{aligned}