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:

${\displaystyle f(x,y)=\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }a_{ij}\phi _{ij}(x,y)}$

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

${\displaystyle a_{ij}={\frac {\langle f(x,y),\phi _{ij}(x,y)\rangle }{\langle \phi _{ij}(x,y),\phi _{ij}(x,y)\rangle }}}$

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