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--- math121a-f23:october_13_friday [2023/10/14 07:59]
pzhou
+++ math121a-f23:october_13_friday [2026/02/21 14:41] (current)
@@ Line 42: / Line 42: @@
 ==== convlution in $x$ space ====
-Convolution is usually denoted as $\star$. Due to our normalization convention, we have the following convolution definition
+Convolution is usually denoted as $\star$.
 If $f$ and $g$ are functions on the $x$ space, then we define
-$$ (f \star g)(x) = (1/2\pi) \int_{x_1} f(x_1) g(x-x_1) dx_1 $$
+$$ (f \star g)(x) =  \int_{x_1} f(x_1) g(x-x_1) dx_1 $$
 If $F$ and $G$ are functions on the $p$ space, then we define
 $$ (F \star G)(p) =  \int_{p_1} F(p_1) G(p-p_1) dp_1 $$
 Fourier transformation sends convolution of functions on one side to simply multiplication on the other side.
-$$ FT(f \star g) = F \cdot G. $$
+$$ (1/2\pi) FT(f \star g) = F \cdot G. $$
 $$ FT(f \cdot g) = F \star G. $$

Lecture Notes