Fourier expansions for basic periodic functions

FOURIER EXPANSIONS FOR BASIC PERIODIC FUNCTIONS (Continued)... 

Fourier Expansions for Basic Periodic Functions The Fourier Transforms Series Expansion Vector Analysis... 

The plane-wave expansion method described above is a general method that applies to any periodic structure. However, using the plane waves as basis functions leads to a large number of Fourier coefficients, and many of these coefficients are related by the symmetry of the system. For an ordered structure, it is possible to reduce the number of independent coefficients by exploiting the point group symmetry of the structure. The basic idea is that, due to the point group symmetry, the Fourier coefficients for the reciprocal lattice vectors within one star are related [26]. A set of new basis functions, which are linear combinations of the plane waves with wave vectors within one star, can be constructed using this observation. Each of these new basis functions is a linear combination of the form. 

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