So far, we have learned many transformations and inverse transformation. Conceptually, we are just trying to decompose a given function $f(x)$ as a linear combination of $e^{ax}$ for various $a$. Because it is an eigenfunction of $(d/dx)$: $(d/dx) e^{ax} = a e^{ax}$

Decomposition of a standing wave with Dirichlet condition. When we pluck a string on a violin, the string vibrate with the endpoints fixed.

Suppose the string's length is $L$, and you took a snapshot of the string and obtained the vertical displacement as a function $f: [0,L] \to \R$ (yes, it is $\R$ valued, not $\C$ valued).

Design a way to do Fourier decomposition of $f$.

If you like, pick $L = 2$ and $f(x) = 1 - |x-1|.$ (just a linear peak).

Damped oscillation, reverse engineering.

Suppose you observe some damped oscillation that goes like $f(t) = A e^{-a t} sin(b t), \quad t > 0$ Can you find the second order equation that $f(t)$ satisfies? Namely, for what constant $A,B$ does $f(t)$ satisfies the equation?

$f“(t) + A f'(t) + B f(t) = 0.$

If you like, pick $a=1, b= 2$.

Tuning the friction.

Consider the pure oscillation equation: $f”(t) + \omega^2 f(t) = 0$ it has solution like $f(t) = a \sin(\omega t) + b \cos(\omega t).$

What happens if we add some friction term? $f“(t) + a f'(t) + \omega^2 f(t) = 0$

• does it matter if $a$ is positive or negative?
• does it matter if $a$ is small or large?

Make a guess first. Then solve it explicitly, for $\omega = 1$, and try various $a$.

Hint: Is there a 'planewave' that solves the equation, i.e. somethign like $e^{ct}$ for some $c$?

You received the following a periodic sequence of numbers,

• sequence of period 9: 0 0 1 0 0 2 0 0 3

Use discrete Fourier transformation to analyse it.

• You can use this input number to as values of $f(x)$ for $x=0,1, \cdots, 8$.
• you know $f(x)$ can be written as

$f(x) = \sum_{p=0}^8 F(p) e^{2\pi i (xp/9)}$ Can you find $F(p)$?

• These $f(x)$ are all real valued, what does that say about $F(p)$? (try to take complex conjugate of the equation)
• Can you guess, which $|F(p)|$s are largest?

If you have done the above exercise, try this one

• sequence of period 10: 2 5 3 8 2 7 1 9 0 4

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?

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' $\Z / N\Z \cong \{0,1,\cdots, N-1\}.$

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

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 $\| F\|_p^2 := \int_\R |F(p)|^2 dp$

$\| f\|_x^2 := (1/N) \sum_{x=0}^{N-1} |f(x)|^2$

Let $F(p)$ be a complex valued function on $p \in \R$, we define $\| F\|_p^2 := \sum_{p=0}^{N-1} |F(p)|^2$

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.

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.

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