Fourier-Transform of an exponential that contains a series I would like to Fourier-Transform the following function:
$\tau\left(x\right) = e^{i\left(H\left(x\right) + G\left(x\right)\right)}$.
$H\left(x\right)$ and $G\left(x\right)$ are Fourier-Series in exponential form, given by
$H\left(x\right) = \sum_{n=-\infty}^{+\infty} a_n e^{inx/g_1}$ and
$G\left(x\right) = \sum_{n=-\infty}^{+\infty} b_n e^{inx/g_2}$.
$G\left(x\right)$ is a known function, so are the corresponding coefficients $a_n$. However, nothing is known about $H\left(x\right)$ and $b_n$ expect that the coefficients are real numbers. $g_1$ and $g_2$ are positive real numbers, $x$ is also real-valued.
I would like to Fourier-Transform $\tau\left(x\right)$ into $T\left(k\right)$. Here I would like to tell which coefficients of $H$ and $G$ contribute to two certain orders of $T\left(k\right)$. 
I tried rewriting $\tau\left( x\right)$ into a product-series, in order to get rid of that "double-exponential". But from this point on, I have no idea how to proceed further.
I realize that things are very easy in case that $\tau\left( x\right) = H\left( x\right) + G\left( x\right)$, without that exponential function.
Online-research did not bring me any further. Is this a well-known problem? Is there a solution?
 A: The characteristic function, https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory), for an  $X_\lambda \in Po(\lambda)$ variable has the form,
$$
e^{\lambda (e^{it} - 1)}
$$ Which looks close to one of your factors. the characteristic function. The characteristic function for $aX_\lambda$ is
$$
  e^{\lambda (e^{i a t} - 1)}
$$ So let's rewrite your formula as
$$
\tau = \exp \left ( \sum a_n(e^{i(nt/g_1 + \pi/2)}-1) + \sum b_n(e^{i(nt/g_2+\pi/2)}-1)\right ) \exp  \left ( \sum (a_n+b_n) \right )
$$
$$
\exp(a_n(e^{i(nt/g_1 + \pi/2)}-1)) = E(\exp(i \frac{n(t + \frac{\pi g_1}{2 n})}{g_1}X_{a_n})) = \phi_{a_n}(t+\frac{\pi g_1}{2 n})
$$ with $\phi_{a_n}$ the characteristic function of $\frac{n}{g_1}X_{a_n}$ Fourier transform of this lead to a complex distribution $Y_n$ of atoms so that the weight is essentially, also note that the fourier transform and the inverse fourier transform is differnt in sign
$$
f(Y = -\frac{n k}{g_1}) = C e^{-a_n}\frac{(i a_n)^k}{k!}
$$ Now the final distribution of this is the convolution of all those distributions.
