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:


 * $$f(x,y) = \sum_{i=1}^\infty\sum_{j=1}^\infty a_{ij}\phi_{ij}(x,y)$$

Where &phi;ij is a 2-dimensional set of orthogonal basis functions. We can define the coefficients as:


 * $$a_{ij} = \frac{\langle f(x,y), \phi_{ij}(x,y)\rangle}{\langle \phi_{ij}(x,y),\phi_{ij}(x,y)\rangle}$$

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