The Fourier series of a function $f: G \to \mathbb C$ where $G$ is a group is the representation of $f$ in terms of characters $\chi_g \in \mathrm{Hom}(G, S^1)$ of $G$.
I understand the case where $G$ is finite and discrete. Now I'm trying to generalise to periodic functions on $\mathbb R$ but I'm not so sure what I'm thinking is correct.
If $f: \mathbb R \to \mathbb C$ is a $[-R,R]$-periodic function, we set $G = \mathbb R / 2R\mathbb Z \cong [0, 2R)$. Then since $f$ is $2R$-periodic we can shift this by $R$ consider $G = [-R, R)$, makes no difference.
In the discrete case $|G|=n$ the characters are $e^{\frac{2 \pi i k}{n}}$ for $0 \leq k \leq n$.
Now for some reason, the characters are $e^{i kx}$ for all $k \in \mathbb Z$. How do I see that these are the characters of $[-R,R)$?