Engineering Analysis/Multi-Dimensional Fourier Series

The concept of the fourier series can be expanded to include 2-dimensional and n-dimensional function decomposition as well. Let's say that we have a function in terms of independent variables x and y. We can decompose that function as a double-summation as follows:

$f(x,y)=\sum _{i=1}^{\mathrm{\infty }}\sum _{j=1}^{\mathrm{\infty }}{a}_{ij}{\varphi }_{ij}(x,y)$

Where φ_ij is a 2-dimensional set of orthogonal basis functions. We can define the coefficients as:

$a_{ij}=\frac{⟨f(x,y),{\varphi }_{ij}(x,y)⟩}{⟨{\varphi }_{ij}(x,y),{\varphi }_{ij}(x,y)⟩}$

This same concept can be expanded to include series with n-dimensions.

Further reading

• Basic Physics of Nuclear Medicine/Fourier Methods discusses using 2D and 3D Fourier reconstruction to get images of the interior of the human body.
• Kevin Cowtan's Book of Fourier: a book of pictorial 2-d Fourier Transforms.

The Feyman lectures Chapter 50 Harmonics