I was wondering about the expansion in a Fourier series of a function,
$$ f(\textbf{r}) = \sum_{\textbf{k}} f_{\textbf{k}} e^{i \textbf{k} \cdot \textbf{r}}, $$
in the context of condensed matter physics. In the theories that discretize the reciprocal space (I only know the Bloch theory, but maybe there are more) with periodic boundary conditions, is the sum in $ \textbf{k} $ only on these vectors? If so, why?
I thought some more about it, and I think I understand it now. Please correct me if I'm wrong.
The Fourier series of a function is given by
$$ f(\textbf{r}) = \sum_{\textbf{k}} f_{\textbf{k}} e^{i\textbf{k} \cdot \textbf{r}}. $$
Imposing periodic boundary conditions, $ f(\textbf{r}) = f(\textbf{r} + \textbf{L}_i) $, where $ \textbf{L}_i $ is the length of the crystal in the direction $ i $, yields
$$ e^{i \textbf{k} \cdot \textbf{L}_i} = 1. $$
Expressing $ \textbf{k} $ as
$$ \textbf{k} = \sum_i x_i \textbf{b}_i, $$
which I think we can do to generate all space if they are linear independent (right?) and using the condition above we have
$$ x_i = \frac{\mathbb{Z}}{N_i}, $$
with $ \textbf{L}_i = N_i \textbf{a}_i $ where $ \textbf{a}_i $ is a Bravais vector, and therefore a vector of the reciprocal space.
