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(Note I have already visited this post but it is about a specific way to calculate the Fourier series of a function, it doesn't adress why we actually can do this). After checking multiple resources, I thought I could use Fourier series to write periodic functions in terms of sines and cosines (or complex exponentiales, same thing), while Fourier transforms can be used to do the same thing but for non-periodic functions. Nonetheless, I realize it's not that simple, since I have just recently learnt we can also write the Fourier series of a non-periodic function if said function is defined in a finite interval. This series would then give us the periodic extension of our function, which I assume we would just then strip back down to our interval of interest.

However, if both periodic and non-periodic functions can be expressed using Fourier series, then what is the Fourier transform good for? Perhaps for the specific case when the interval is non-finite and the function is non-periodic?

Lagrangiano
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    The purpose of the concept of periodic vs. non-periodic functions is to be able to talk about functions with infinite domain. A function defined only on a finite interval is not "non-periodic." In fact, according to the usual definition of periodic functions, any function on an interval is periodic; just set the period to at least the length of the interval. – David K Apr 13 '24 at 18:00
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    Precisely, the technique of Fourier series is to take a part of a function like a DNA braid, then copy it again and again making it periodic (in the even case) – Jean Marie Apr 13 '24 at 19:41
  • @JeanMarie nice analogy! Is there a similar one for the odd case? – Lagrangiano Apr 13 '24 at 20:47
  • The Fourier series of a non-periodic function is always periodic whether the function is even, odd, or neither. The coefficients of a Fourier series are derived from a Fourier transform where the integration range is truncated to the interval of interest, and the length of this interval determines the period of the Fourier series. Your question seems to overlap this question. – Steven Clark Apr 13 '24 at 22:59
  • See my answer at https://math.stackexchange.com/questions/3128420/fourier-series-of-non-periodic-function-fx-e-fracaxl?rq=1 – Jog Apr 13 '25 at 10:25

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