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--- math121a-f23:october_13_friday [2023/10/14 00:28]
pzhou [FT Conventions]
+++ math121a-f23:october_13_friday [2023/10/14 01:13] (current)
pzhou [convlution in $x$ space]
@@ Line 8: / Line 8: @@
  $F(p) = (1/2\pi) \int_\R f(x) e^{-ipx} dx.$
-Discrete Fourier transformation (OK, I switched to Boas convention)
+Discrete Fourier transformation
-Fix a positive integer  $N$ .  $x,p$  are valued in $\Z / N\Z \cong \{0,1,\cdots, N-1\}$.
- f(x) = \sum_{p \in \Z / N\Z} F(p) F(p) e^{2\pi i \cdot px/N}.
+Fix a positive integer  $N$ .  $x,p$  are valued in the 'discretized circle'
- F(p) = (1/N) \sum_{p \in \Z / N\Z} f(x)  e^{-2\pi i \cdot px/N}.
+$$ \Z / N\Z \cong \{0,1,\cdots, N-1\}.
+ $f(x) = \sum_{p \in \Z / N\Z} F(p) e^{2\pi i \cdot px/N}.$
+ F(p) = (1/N) \sum_{x \in \Z / N\Z} f(x)  e^{-2\pi i \cdot px/N}.
+==== Norm in the Continous Fourier transformation ====
+Let  $f(x)$  be a complex valued function on  $x \in \R$ , we define
+ $\| f\|_x^2 := (1/2\pi) \int_\R |f(x)|^2 dx$
+Let  $F(p)$  be a complex valued function on  $p \in \R$ , we define
+ $\| F\|_p^2 := \int_\R |F(p)|^2 dp$
+==== Norm in the Discrete Fourier transformation ====
+ \| f\|_x^2 := (1/N) \sum_{x=0}^{N-1} |f(x)|^2
+Let  $F(p)$  be a complex valued function on  $p \in \R$ , we define
+ \| F\|_p^2 := \sum_{p=0}^{N-1} |F(p)|^2
+==== Parseval Equality ====
+If  $F(p)$  is the Fourier transformation of  $f(x)$ , then  $\|F\|^2_p = \|f\|^2_x.$
+We proved in class the discrete case. The continuous case is similar in spirit, but harder to prove.
+===== Convolution =====
+Consider two people, call them Alice and Bob, they each say an integer number, call it a and b. Suppose  $a$  and  $b$  both have equal probability of taking value within $\{1,2,\cdots, 6\} $, we can ask what is the probabity distribution of$ a+b$?
+We know  $P(a=i) = 1/6$ ,  $P(b=i) = 1/6$  for any $i=1,\cdots, 6$, otherwise the probabilit is 0.  Then
+ $P(a+b = k) = \sum_{i+j=k} P(a=i) P(b=j).$
+This is an instance of convolution.
+==== convlution in  $x$  space ====
+Convolution is usually denoted as  $\star$ .
+If  $f$  and  $g$  are functions on the  $x$  space, then we define
+ $(f \star g)(x) =  \int_{x_1} f(x_1) g(x-x_1) dx_1$
+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.
+ $(1/2\pi) FT(f \star g) = F \cdot G.$
+ $FT(f \cdot g) = F \star G.$

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