I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in the classical DFT for complex numbers).
First, in the classical setting, sums of the type $$ A(N) = \sum_{k=0}^{\infty} a_k c_N(k) $$ are called Ramanujan expansions, where $c_N$ is a Ramanujan sum. See here:
https://en.wikipedia.org/wiki/Ramanujan%27s_sum#Ramanujan_expansions
There's a lot of research on determining when does the sum converge, and if so, given $A$, it is interesting to obtain the Ramanujan/Fourier coefficients $a_k$. Admittedly, there's another variation of the expansion where the sum runs over the modulus $N$, but let's focus on the one above for the moment.
Now let $\mathbb{F}_q$ be a finite field with characteristic $p$ and let $N \mid (q-1)$. It is known that $c_N(k) \in \mathbb{Z}$, so reducing modulo $p$ we can view $c_N(k)$ as lying in $\mathbb{F}_p \subseteq \mathbb{F}_q$. Next, convergence of infinite sums of elements in a finite field is problematic, so we take one period of the terms in the sum above. $$ A(N) = \sum_{k \in \mathbb{Z}_N} a_k c_N(k), $$ where each $a_k \in \mathbb{F}_q$.
Would this be a good analogy? For instance, given a function $A(N)$, can we have a way of determining the coefficients $a_k$? Are these unique? As an example, can we get an explicit expression for the coefficients $a_k$ here: $$ \phi(N) = \sum_{k \in \mathbb{Z}_N} a_k c_N(k)? $$ $\phi$ is the Euler's totient function.
BTW I've had trouble finding introductory material online on classical Ramanujan expansions. It would be nice if someone could recommend me something nice to read. Thanks!