For a function $g$ defined at discrete points $x_n = n a$ with $n \in \{0, \ldots, N\}$ with periodicity $g(x_0) = g(x_N)$ the discrete Fourier transformation reads
$g(x_n) = \frac{1}{Na} \sum_{q \in 2 \pi \mathbb{Z}/Na} \tilde{g}(q) \mathrm{e}^{ \mathrm{i} q x_n} $
with inverse transformation
$\tilde{g}(q) = a \sum_{i=1}^N g(x_n) \mathrm{e}^{ \mathrm{i } q x_n}$.
Now from $\tilde{g}(q) = \tilde{g}(q + 2 \pi m /a)$ for $m \in \mathbb{Z}$ one should be able to see, that the expression above simplifies to
$g(x_n) = \frac{1}{Na} \sum_{q_m} \tilde{g}(q_m) \mathrm{e}^{ \mathrm{i} q_m x_n} $ with $q_m = 2 \pi m / Na$ and $m = 0, \ldots N-1$.
I am having trouble making this conclusion and would be grateful if someone could help.