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

## DefinitionEdit

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

## The translation theoremEdit

Given a fixed position vector **R**_{0}, if *g*(**r**) = *ƒ*(**r** − **R**_{0}), then

Now, change **r** to a new variable by:

## The convolution theoremEdit

The convolution of *f* and *g* is usually denoted as *f*∗*g*, 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

- .