====== October 13, Friday ======
Parseval Equality says, Fourier transformation, as a linear map from one function space (function on x), to another function space (function on p), preserves 'norm'. Norm is just a fancy way of saying 'length of a vector'. 

What do we mean by the length of a function? 
==== FT Conventions ====
Continuous Fourier transformation (OK, I switched to Boas convention)
$$ f(x) = \int_\R F(p) e^{ipx} dp. $$
$$ F(p) = (1/2\pi) \int_\R f(x) e^{-ipx} dx. $$

Discrete Fourier transformation

Fix a positive integer $N$. $x,p$ are valued in the 'discretized circle'  
$$ \Z / N\Z \cong \{0,1,\cdots, N-1\}.$$ 

$$ f(x) = \sum_{p \in \Z / N\Z} F(p) e^{2\pi i \cdot px/N}. $$
$$ F(p) = (1/N) \sum_{x \in \Z / N\Z} f(x)  e^{-2\pi i \cdot px/N}. $$

==== 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. 
$$ (1/2\pi) FT(f \star g) = F \cdot G. $$
$$ FT(f \cdot g) = F \star G. $$