Is there an analog of Fourier series in the function field setting based on the Carlitz exponential? I mean, something like:


Let $\Omega$ be the completion of an algebraic closure
 of $\mathbb F\_q\left(\left(\frac1T\right)\right)$ and $\exp\_{\mathcal C}(z)=\sum\_{n\ge0}\frac{z^{q^n}}{D\_n}$ be the Carlitz exponential on $\Omega$ where $$D\_n=\prod\_{\substack{a\in\mathbb F\_q[T]\\ \deg a= n\\ a\text{ monic}}}a.$$
One denotes by $\xi\in\Omega$ the smallest (relatively to degree) period of $\exp\_{\mathcal C}$. Let $f$ be an entire function on $\Omega$ such that $f(z+a)=f(z)$ for all $z\in\Omega$ and $a\in\mathbb F\_q[T]$. Can we expand $f$ as
$$f(z)=\sum\_{a\in\mathbb F\_q[T]}b\_a\exp\_{\mathcal C}(\xi a z),\text{ where }b\_a\in\Omega.$$



Fourier expansions in function fields
Yes, there is and is called $t$-expansion rather than $q$-expansion - may be, though, you might want to replace your $\Omega$ by
$$
\widehat{\bar{K\_\infty}}\setminus K\_\infty
$$
where I denote by $K\_\infty$ your completion $\mathbb{F}\_q\big(\big(\frac{1}{T}\big)\big)$. This is the content of the paper *On the coefficients of Drinfel'd modular forms* by E-U. Gekeler, Inventiones Math. 93 (1988).


