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In Fourier Analysis, we can reconstruct a function $f(x)$ from its Fourier transform $\mathcal{F}(\omega)$ by applying the inverse Fourier transformation. This led me to pose the following question:

Given the Fourier coefficients $a_0$, $a_n$, $b_n$ and the period length of some periodic function, is it possible to reconstruct the function $f(x)$?

My initial approach quickly came to a halt as I simply tried to rewrite the coefficient equations as ODEs for $f(x)$. Knowing however how we arrive at the fourier transform from the discrete series, I can't help but wonder (or hope) that there exists some nice property of Fourier series which allows this inversion - or more generally, under which conditions can a function be reconstructed given its definite integral over some interval?

And if not, which additional feature of the continuous case makes it possible to perform this operation?

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  • $\begingroup$ Do you mean like $$f(x) = a_0+\sum_k \left(a_k\cos k\omega x + b_k\sin k\omega x\right)$$ for some $\omega$ derived from the period? $\endgroup$ – peterwhy Jun 22 at 19:05
  • $\begingroup$ yes, given the equations for the coefficients in terms of $n$ and $\omega$, reconstruct $f(x)$ in terms of $x$. $\endgroup$ – LoschmidtsSchnitzel Jun 22 at 19:10
  • $\begingroup$ related: math.stackexchange.com/questions/32692/… $\endgroup$ – LoschmidtsSchnitzel Jun 22 at 20:16

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