Differences

This shows you the differences between two versions of the page.

--- math121a-f23:october_13_friday [2023/10/14 01:00]
pzhou
+++ math121a-f23:october_13_friday [2023/10/14 01:13] (current)
pzhou [convlution in $x$ space]
@@ Line 43: / Line 43: @@
 ==== convlution in  $x$  space ====
 Convolution is usually denoted as  $\star$ .
-Due to our normalization convention, we have some funny  $1/2\pi$  factor there. Again, this may differ from conventions in other places.
 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