/fourier analysis/

[Anonymous No.11295375]

Please teach me the fourier series/FT. I am familiar with every single detail of it separately, but I can't connect the dots. Watching youtube videos just confused me further. You my friends are my last hope.

[Anonymous No.11295385]

>Fourier series
DUDE
GEOMETRIZATION OF ANALYSIS
DUUUDE
ORTHONORMAL BASIS OF [math]L^2(a, b)[/math]
>Fourier trasnform
DUDE FREQUENCY DOMAIN
CONTINUOUS FOURIER SERIES
INTEGRAL TRANSFORM

[Anonymous No.11295947]

Every periodic function can have a Fourier series, either in the cosine and sine form of in the complex exponential form.
It's pretty much using sine and cosine functions with an integer frequency as a base of the space of periodic functions.
The Fourier Transform is pretty much the same thing but with real continuous frequencies (you can find Fourier Series that way too).
You can compute the Fourier Transform of a fuckload of functions as long as they are L2 or L infinite but you would need distributions for that.
Is there anything in particular you're struggling with OP ?

[Anonymous No.11296043]

Make sure you have a solid grasp of Linear Algebra, particularly Vector spaces. Do a bunch of problems involving Fourier analysis.
And if desperate, watch the 3blue1brown videos. They are helpful for some.

[Anonymous No.11296488]

>>11295947
I am struggling with a couple of things:
1. Motivation of the derivation: FS is compared to the Taylor Series, but "we can't differentiate so we have to integrate". I don't understand why we would want to integrate or differentiate and what our goal is with that derivation, such as what do we want to show?
2. I don't understand why orthogonality is important. I kinda know what that is but somehow that doesn't help me. I know at some point in the derivation some parts get cancelled out because of it.

>>11296043
>3blue1brown
oh nooo, I watched his video on FT with a bunch of circles and wanted to kill myself. He was just too excited drawing those circles and I was drowning in them.

[Anonymous No.11296550]

>>11296488
iirc from the derivation they use the orthogonality is used for calculation of the coefficients of each term in the Fourier series. The orthogonality makes the inner product trivial, which is used to simplify an expression for each coefficient.

[Anonymous No.11296780]

>>11296488
>They are helpful for some.
Bunch of people in my class didn't get FFT until they watched his video, which stupid as it might be, helped people understand them.
There's many ways to kill a cat.

[Anonymous No.11296782]

>>11295385
have sex

[Anonymous No.11297772]

>>11296780
Sure but "getting it" may mean different things to different people. Those videos are just a different way of visualizing a complex signal that consists of harmonics. Only instead of the cartesian plane, they animate a bunch of sticks rotating in the polar plane with different amplitudes, phases and frequencies and they also trace some fancy shapes. Thats cool and all but totally unnecessary and kinda distracting. And also misleading since all of these vectors should originate from 0,0 but they sit on top of which other for the cooler effect. And still doesn't explain WHY FT works or how to derive it.

[Anonymous No.11297806]

>>11296488
Are you familiar with differential equations?
If so, you should know about the superposition principle. Basically any solution that can be fed through the equation is added into the "general solution", and then you can plug in values to find particular solutions.

If you're not already familiar with this, you need to go learn that stuff first because Fourier series is a tier above this.

Anyway, there are particular differential equations which we have methods of solving, such as the Heat Equation and Laplace's Equation.
Let's take the Heat Equation. We find that the general solution involves a sum of sines and cosines. And since we can plug in ANY boundary values into this general solution, we find that sines and cosines can sum together to approximate ANY function.

I am not totally certain, but I strongly believe this is where the idea was derived from.

There is more I can go into if you want. But I'm not sure you're even still in this thread

[Anonymous No.11297821]

>>11296782
Dilate.

[Anonymous No.11297871]

>>11297806
Thanks, any insight is much appreciated.

Yes I am still in this thread.
Re DEs, I only know the basic concepts: separation of variables, move them to the opposite sides, then integrate. I am also familiar with simple 1st order equations: simple RC/RL circuits that result in y' + y = 0. I am supposed to be familiar with the 2nd order equations (RLC circuits or spring systems) by now, but I am kinda falling behind.
I looked at the Heat equation and it's way over my head. How should I work up to it?
Regarding a general solution vs particular solutions I think I read up on that. In the EE context it means a forced response vs natural response or something like that. But I am not sure it is directly related to the Fourier transform. I know that more complicated stuff calls for the Laplace transform but I don't want to get there yet since it is even more general and more complicated. I know a bunch of individual facts but I can't line them up and connect the dots to have a clear picture.

[Anonymous No.11297905]

>>11297871
I believe they go the idea for the Laplace Transform from the Fourier Transform. They hammered it out in their math rooms and figured out that it actually works. It's an interesting technique.

Anyway, how would you solve this?
[math]A\frac{\partial^2 y}{\partial x^2} + B\frac{\partial y}{\partial x} + Cy = f(x)[/math]
Where A, B, and C are arbitrary constants?

Spoiler alert, you basically just plug in exponentials with a certain power (like [math]e^{\lambda x}[/math]) and then solve for your [math]\lambda[/math]'s.
There is some more subtlety than that and you should read up on it, but that's the general idea.

As for nonlinear differential equations (which would involve terms like y''^2) there are sometimes ways to get solutions, but usually not. These are generally classified "unsolvable", although there are ways to approximate solutions in certain domains. Getting these approximations is often part of the task of a physicist or engineer.

Ok, so how does this help us with the Heat Equation? This is a PDE
On the left we have a term which is a derivative with respect to space, on the right we have a term which is a derivative with respect to time. We simply assume they equal a constant (there are good reasons to make this assumption), and then solve the two separately. And we get sines and cosines out.

You probably won't understand what I've been saying, and that's because you need to learn about techniques to solve Differential Equations. That will lead you to Fourier series naturally

[Anonymous No.11298069]

>>11297905
>I believe they go the idea for the Laplace Transform from the Fourier Transform
I think that's right, but the FT is still a specific kind of LT. LT is more general. Quality post, btw.
>dude jumping into jω domain
>dude frequency response
>dude muh catch-all LTI system solver

[Anonymous No.11298086]

>>11297772
>And also misleading since all of these vectors should originate from 0,0 but they sit on top of which other for the cooler effect

>doesnt know how to add vectors together

[Anonymous No.11298270]

Not OP, but I'm trying to learn about atoms in L^1. The most basic result is showing that H_r ^1 is complete, with the norm to be the infimum over the infinite linear combinations of atoms. I tried mimicking the proof that L^p is complete, but got stuck at the very end because I'm not sure if I have an analog of the DCT from the L^p setting restricted to the case of H_r ^1. Can any anons help?

Pic related, using functional Analysis right now