Chemical Dynamics/Electrostatics/Fourier Transforms

The Fourier transform is a useful mathematical transformation often utilized in many scientific and engineering fields. Here we extract useful concepts of Fourier transformation and logically arrange them to form a foundation for the Ewald summation and other related methods in electrostatics. Readers could check out other more mathematically formal introduction of Fourier transform

Definition edit

We use the following convention in which the Fourier transform is a unitary transformation on the 3-D Cartesian space R3, the Fourier transform and its inverse transform are symmetric:

 
 

The translation theorem edit

Given a fixed position vector R0, if g(r) = ƒ(r − R0), then  

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \hat{g}(\mathbf{k})= e^{- i \mathbf{k}\cdot \mathbf{R}_0 }\hat{f}(\mathbf{k}).}

Proof
 
 
 

Now, change r to a new variable by:  

 
 
 
 

The convolution theorem edit

The convolution of f and g is usually denoted as fg, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted:


 


The convolution theorem for the Fourier transform says:

If

 

then

 .
Proof