If you have any questions, please ask. I love yapping.
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Let's say we have some periodic function f. The goal is to represent it as a bunch of sine waves. This is called a Fourier series (FS).
The FS coefficients are given by
⌠2⋅π
1 ⎮ -i⋅k⋅x
f̂(k) = ─── ⎮ f(x)⋅e dx
2⋅π ⌡0
for whole numbers k. We can then approximate the original function with
n
___
╲ -i⋅k⋅x
╱ f̂(k)⋅e
‾‾‾
k = -n
and make the approximation better by making n larger.
For non-periodic functions, we can still get an approximation! The Fourier Transform (FT) uses a continuous range of frequencies instead of whole numbers, and the sum is replaced by an integral (the inverse FT).
FT's are used everywhere, for example in signal processing and spectroscopy. If a recording contains an annoying high-pitched noise, we can apply the FT and filter the unwanted frequencies. Spectroscopy separates light into its different wavelengths. By comparing the resulting spectral lines with known measurements, we can determine which elements and molecules are present in stars and planets, for example.
idk what more to add ._.
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This reminds me of a story: when my first girlfriend broke up with me I was devastated. My favorite past time the next year was to get drinky, use music and internet threads to make me sad instead of numb, and then pour through our entire relationship's instant messenger chat logs to see if I could understand where it all went wrong. Finally, I realized how unhealthy it all was and I put every log, every picture, and some other things on a jump drive and gave it to my best friend to keep safe for me until I could be trusted with it again. A couple years after that, I asked them to throw it away, I had finally dug myself out of that hole I dug and moved on. They had lost it XD
I don't know when I ever would've gotten better had I not put my self-harm material out of reach. It didn't stop every thought, and didn't stop a particularly toxic pattern I had with my ex where she would call whenever her life was falling apart because I always put my own emotional health on hold to be there for her, but it did keep me from my worse self-inflicted rumination pattern.
(This section isn't toward you SuperZutsuki, but to anyone reading) I don't know how we do that with climate change, negative self talk, the Palestinian genocide, rising fascism, or any of the other nightmares we live with, but if there is a way to put it away, put it into someone you trust's hands until you need it again, try. See if you can take a week of social media, with a friend who will let you know if anything really important happens, if the week goes well maybe make it a month. Find hopeful media to engage with, not bullshit saccharine media, but media where positive seeds are planted and cared for. And maybe most importantly, even though its incredible, awfully frustrating at times, connect with people off line in the work to make the world just a bit better. There are countless ways to do this, labor unions, mutual aid groups, food not bombs orgs, urban farms, communist and/or anarchist groups, etc. You have to be not alone more times than at work if you're ever going to be okay, and you can't be behind a mask then (I mean identity mask, not COVID mask, so you have to be out as your true gender, your real neurotype, etc.) or it won't work. But, baby steps of getting back into the world...see what your public library has available, you might be surprised what kinda things you can do for free.