Suppose we know the Fourier series of a function $$f(x) = \sum_k c_k e^{ikx}$$. Is there any explicit relation to find the fourier series of $\frac{1}{f(x)}$ in terms of the coefficients of the fourier series of $f(x)$? i.e, if the fourier series of $\frac{1}{f(x)}$ is $$\frac{1}{f(x)}= \sum_k d_k e^{ikx}$$can we express $d_k$ in terms of $c_k$'s?
(Assuming $f(x)$ and $\frac{1}{f(x)}$ both exist throughout $\mathbb{R}$ and are smooth.)
No. Not much more to say - can't prove there is no such formula, but there isn't.
For example: Recall Wiener's theorem: If $f=\sum a_ne^{int}$, $f$ has no zero on the circle, and $\sum|a_n|<\infty$ then $1/f=\sum b_ne^{int}$, where $\sum|b_n|<\infty$. If there were a formula for $b_n$ in terms of $a_n$ people would prove this, or try to do so, using that formula. It's precisely the non-existence of such a formula that makes the theorem amazing: Somehow we show that $\sum|a_n|<\infty$ implies $\sum|b_n|<\infty$ even though we have no explicit way to get our hands on $b_n$ given $(a_n)$.