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Consider a cell periodic function, , which may be written in terms of its discrete Fourier series

(50) 
Consider also the bandwidth limited version of this same function,
, which has only the same frequency components as :

(51) 
It can be shown that the projection of onto a particular basis function is exactly equal to that of
, and that furthermore, replacing the integral by a discrete sum over grid points leads to exactly the same answer:
This result is very useful for our purposes as it tells us that the overlap integral of any cell periodic function with a function that is represented by our basis set, can be evaluated exactly as a summation over grid points.
Next: Basis for the fine
Up: Basis Set
Previous: Localisation and Orthogonality
Peter D. Haynes
20021029