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 $$f(z)=\sum_{a\in\mathbb F_q[T]}b_a\exp_{\mathcal C}(\xi a z),\text{ where }b_a\in\Omega.$$