Appendix: properties of the Fourier transform


The Fourier transform of a function f(r) is defined by

F(q) = FT{ f }(q) = ò  f(r) exp(iqr) d3r
The Fourier transform can be inverted:
f(r) = FT-1{ F }(r) = ò F(q) exp(-iqr) d3q / (2p)3
In the above considerations, we often omitted the factor (2p)3 for the sake of clarity.

The Fourier transform is linear :

FT{ a f + b g} = a FT{ f } + b FT{ g }
for two functions f(r) and g(r) and for complex numbers a, b. A shift of a function in real space is converted into a phase factor in q space:
FT{ f(r-r0) } = FT{ f(r) } exp(i qr0)


Convolution theoreme

A very important application of the Fourier transform (FT) is the convolution theoreme. A convolution of two functions f(r) and g(r) is defined by

{f * g}(r) = ò f(r-r') g(r') d3r'
The convolution theoreme states that the FT of a convolution is simply the product of the FT's of the functions:
FT{ f * g } = FT{ f } FT{ g }
There is also an inverse form of the convulution theoreme:
FT { f g } = FT{ f } * FT{ g }
i.e. the FT of a product of two functions is the convolution of the FT's of the functions.
 

Fourier transforms of special functions

a) d-function

FT{ d(r) } = 1
follows trivially from the Fourier integral.

b) unity function: u(r) = 1

FT{ u } = d(r)
This is equivalent to the Fourier representation of the d-function

c) Heavyside function f(z) = 1, z<0; 0 else.

The 1D FT is given by

FT{ f(z) } = i/qz
In order to represent an interface in 3D we write:
w(x,y,z) = u(x) u(y) f(z)
and the Fourier transform is
FT{ w } = d(qx) d(qy) (i/qz)
d) lattice function

latt(r) = Smnp d(r-rmnp)
with rmnp = ma + nb + pc yields the lattice function of the reciprocal lattice
Latt(q) = Skhl d(q-ghkl)
 

e) 1D Gaussian

gauss(z) = (2p s)-1/2 exp( -z2/ 2s2 )
is again a Gaussian in reciprocal space with the width reciprocal to the width in real space:
Gauss(qz) = exp(- s2 qz2)


f) cut-off function  rect(z/d) = 1, |z/d| < 1; 0 else

FT{ rect } = sin(d qz) / (d qz)