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--- math121a-f23:october_13_friday [2023/10/14 07:32]
pzhou [Norm in the Discrete Fourier transformation]
+++ math121a-f23:october_13_friday [2026/02/21 14:41] (current)
@@ Line 28: / Line 28: @@
 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.
+$$ (1/2\pi) FT(f \star g) = F \cdot G. $$
+$$ FT(f \cdot g) = F \star G. $$

Lecture Notes