It is well known that if $f$ is a real analytic function, then its Fourier coefficients decay exponentially. That is, for a meromorphic $f$ say, if $$f(z)=\sum_{n\in\mathbb{Z}}c_n e^{inz},$$ then there are constants $K$ and $q$ such that $$c_n=Kq^{n}(1+o(1)), \quad n\to\pm\infty.$$
In the following assume that, for some $r>0$, $f$ is analytic in $|z|<r$ and it has a singularity (a simple pole, for instance) at one point $z_{0}$ lying on the circle $|z|=r$. I would like to know how exactly constants $K$ and $q$ can be deduced from $f$. I realize that it is related somehow with the singularity $z_0$ and corresponding residuum of $f$ at $z_0$. So far, I found only the Darboux's method theorem which is exactly what I need but for power series, not Fourier series.
Can somebody explain this? Is there some reformulation of the Darboux's method for Fourier series in literature? Many thanks.
Exponential decay does not mean that $$c_n=Kq^{n}(1+o(1)), \quad n\to\pm\infty.$$ It means $c_n=O(q^n)$ for some $q\in (0,1)$. There is no $K$ in general.
If a function is holomorphic on $|z|<r$ for some $r>1$, then its Fourier coefficients on the unit circle are the same as its Taylor coefficients at $0$. The latter satisfy $c_n = O(q^{n})$ with $q=1/r$, and this is the smallest possible $q$ here. This is just a restatement of the fact that $$1/r = \limsup |c_n|^{1/n}$$