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I understand how to calculate the fourier coefficients and I understand the importance of orthogonality of sines and cosines. But why can any periodic functio be expressed as a linear combination of sines and cosines? Fourier series is such a common and important subject in math, but I can not find a clear explanation why it is possible?

The Taylor expansion somehow makes sense, as you use derivatives to calculate the function. But I don't see how the fourier series work. Could someone explain this clearly in not too opaque theorems?

bananenheld
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    Have you tried the video by 3Blue1Brown? – Hans Lundmark Apr 23 '23 at 14:55
  • This doesn't explain why it is possible, just that it is possible – bananenheld Apr 27 '23 at 09:40
  • OK... Then maybe you could try to clarify what exactly you're after when you want to know how Fourier series “work”? Are you looking for proofs or for more intuitive/philosophical explanations? – Hans Lundmark Apr 27 '23 at 18:07
  • So my point is that in most texts about fourier series it is just a given that any function can be expanded as a fourier series. But why is this so, why can you do this? Maybe fourier series don't converge to its function at all? How is this discovered, what is an intuitive explanation? – bananenheld Apr 29 '23 at 11:18
  • A vibrating string, with its fundamental frequency and overtones, provides pretty good intuition for why a periodic wave can be expressed as a sum of sine waves, doesn't it? But of course it's not at all obvious whether you can do it for all periodic functions, or what exactly the conditions are, and so on. The study of Fourier series has kept mathematicians busy for a long time, and has actually been the source of many important and deep developments in mathematics (such as Cantor's set theory, to mention just one example). – Hans Lundmark Apr 29 '23 at 14:50
  • Thank you for your reaction but how does this answer my question. – bananenheld Apr 29 '23 at 17:44
  • I suppose I still don't understand what your question actually is then. – Hans Lundmark Apr 30 '23 at 05:09
  • If you want the details of why (and in what sense, and under which assumptions) the Fourier series of $f$ converges to $f$, read the proofs in any decent textbook. If you're wondering how all that theory about Fourier series was developed, what I'm trying to say is that it was quite complicated and took decades of hard work by many brilliant mathematicians, so don't expect any simple explanation. But if you just want some intuition for how one might get the idea to expand in sines and cosines, consider for example the vibrating string (and surely this is also explained in any decent textbook). – Hans Lundmark Apr 30 '23 at 05:09

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