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Always blew my mind that every signal can be recreated simply by adding different sine waves together.
Back in my uni days I did not get why that works. Why are sine waves special? Turns out... they are not! You can do the same thing using a different set of functions, like Legendre polynomials, or wavelets.
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This kind of statement is too vague to be useful. I even know all the words in there and I still don't know what you mean.
Ask away.
Always blew my mind that every signal can be recreated simply by adding different sine waves together.
Only if it is band-limited.
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Hmm I always viewed it as sum over orthogonal bases. Sine/Cosine is choice, while there can be many other bases, like Legendre polynomials. Can you explain more how FT is linked to 'a sum over equivalence classes'?
In the case of the Fourier transform, it maps from time domain to frequency domain. In the frequency domain, we can see the amplitude (count) of the signal at each frequency.
3Blue1Brown made a video with great visualizations: https://www.youtube.com/watch?v=spUNpyF58BY
To be extra clear: it’s a very good video and you should watch it if you don’t have a feel for the Fourier transform. I’m just trying to proactively instil a tiny bit of dissatisfaction with what you will know at the end of it, so that you will then go looking for more.
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Your earlier post comes off as link-spam, as it uses general language to connect something tangentially related to what you're interested in to drive traffic to your website. This is not conducive to questions.
You can still ask.
Earlier quoted context omitted.
You can still ask.
I know and use wavelet, Fourier, and other transforms on a daily basis, but your post did not stimulate any interest in me. Also, your reply on the other thread does not strike me as particularly insightful (I'd argue that your 'frequency decomposition' interpretation is flat-out wrong).