# Reconstructing a function given its Fourier coefficients

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?

• 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? – peterwhy Jun 22 at 19:05
• yes, given the equations for the coefficients in terms of $n$ and $\omega$, reconstruct $f(x)$ in terms of $x$. – LoschmidtsSchnitzel Jun 22 at 19:10
• – LoschmidtsSchnitzel Jun 22 at 20:16