Multiplication of fourier series

[mordechai9]

Say you have two functions, F(x,y), and G(x,y), and you want to expand them in finite Fourier series. Let their coefficients be designated as F_ij and G_ij. When you multiply the two functions, you get X=FG, and this should also have its own Fourier series, call its components X_mn. What is the relation between F_ij, G_ij, and X_mn?

I was hoping you had something like X_ij = F_ij G_ij, but I've been looking at this for a little while and it seems you don't have any nice relation like that.

 

[Cody Palmer]

Check out the Cauchy product, which has to do with multiplying series.. Wikipedia has a good article on it http://en.wikipedia.org/wiki/Cauchy_product"

 

[elibj123]

In the case of "two dimensional sequences" you'll have:$$X_{m,n}=\sum_{j=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}F_{m-j,n-k}G_{j,k}$$

This, by the way, resembles the convolution theorem of the Fourier Transform, and actually the above operation between two sequences (in this special case they are two dimensional) is a discrete convolution.