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authorPrefetch2024-09-29 22:15:59 +0200
committerPrefetch2024-09-29 22:15:59 +0200
commitfda947364c33ea7f6273a7f3ad1e8898edbe1754 (patch)
tree2c393568e58c100b3fad465f23a17089eb3a06d7 /source/know/concept/convolution-theorem
parent766d05aac6f701fa85b7ceed1ce3a473a62cae55 (diff)
Improve knowledge base
Diffstat (limited to 'source/know/concept/convolution-theorem')
-rw-r--r--source/know/concept/convolution-theorem/index.md8
1 files changed, 4 insertions, 4 deletions
diff --git a/source/know/concept/convolution-theorem/index.md b/source/know/concept/convolution-theorem/index.md
index 3f9eafb..8462fcc 100644
--- a/source/know/concept/convolution-theorem/index.md
+++ b/source/know/concept/convolution-theorem/index.md
@@ -24,10 +24,10 @@ and $$A$$ and $$B$$ are the constants from its definition:
$$\begin{aligned}
\boxed{
\begin{aligned}
- A \cdot (f * g)(x)
+ A \: (f * g)(x)
&= \hat{\mathcal{F}}{}^{-1}\Big\{ \tilde{f}(k) \: \tilde{g}(k) \Big\}
\\
- B \cdot (\tilde{f} * \tilde{g})(k)
+ B \: (\tilde{f} * \tilde{g})(k)
&= \hat{\mathcal{F}}\Big\{ f(x) \: g(x) \Big\}
\end{aligned}
}
@@ -45,7 +45,7 @@ $$\begin{aligned}
\\
&= A \int_{-\infty}^\infty g(x') \: f(x - x') \dd{x'}
\\
- &= A \cdot (f * g)(x)
+ &= A \: (f * g)(x)
\end{aligned}$$
Then we do the same again,
@@ -59,7 +59,7 @@ $$\begin{aligned}
\\
&= B \int_{-\infty}^\infty \tilde{g}(k') \: \tilde{f}(k - k') \dd{k'}
\\
- &= B \cdot (\tilde{f} * \tilde{g})(k)
+ &= B \: (\tilde{f} * \tilde{g})(k)
\end{aligned}$$
{% include proof/end.html id="proof-fourier" %}