In this chapter we review the definitions and some properties of Fourier transforms. We first treat one-dimensional non-periodic functions f( x) with Fourier transform F( k), the domain of both coordinates x and k being the set of real numbers, while the function values may be complex. The functions f and F are piecewise continuous and exists. The domain of x is usually called the real space while the domain of k is called reciprocal space. Such transforms are applicable to wave functions in quantum mechanics. In Section 12.6 we consider Fourier transforms for one-dimensional periodic functions, leading to discrete transforms, i.e., Fourier series instead of integrals. If the values in real space are also discrete, the computationally efficient fast Fourier transform (FFT) results (Section 12.7). In Section 12.9 we consider the multidimensional periodic case, with special attention to triclinic periodic 3D unit cells in real space, for which Fourier transforms are useful when long-range forces are evaluated.

12.1 Definitions and Properties

The relations between f( x) and its Fourier transform (FT) F( k) are

The factors 1 / are introduced for convenience in order to make the transforms symmetric; one could use any arbitrary factors with product 2 ?. The choice of sign in the exponentials is arbitrary and a matter of convention. Note that the second equation follows from the first by using the definition...