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