Discrete Fourier transforms

• In practice, we sample $u_{\bf k}({\bf r})$ discretely on a uniform grid of $N$ points $\{ {\bf r}_n \}$:
  $$\begin{eqnarray*} u_{\bf k}({\bf r}_n) &=& \sum_{\bf G} c_{\bf k}({\bf G}) \exp(... ..._n} u_{\bf k}({\bf r}_n) \exp( -{\rm i} {\bf G} \cdot {\bf r}_n) \end{eqnarray*}$$
  
  
  
  
  

• The sum over reciprocal lattice vectors ${\bf G}$ now only runs over those below the Nyquist frequency determined by the real-space grid.
• Although these results are an approximation to the continuous Fourier series, the above inversion theorem is exact.