Use of FFT in the multiplication of multinomials

I'm aware that one can use a Fast Fourier Transform (FFT) to take the cost of multiplication of two polynomials of degree N from O$(N^2)$ to O$(N \ln N)$ (which is an amazing reduction when dealing with large polynomials!). Does a similar transformation procedure exist for multinomials?

I'm interested in the special case where the number of independent variables is only two, ie. $h(x,y) = f(x,y)g(x,y)$, but I'd love to read up on the general procedure.

Solution 1:

Community wiki answer so the question can be resolved:

As pointed out in the comments, this can be done using multidimensional FFT, with the exponents of the variables serving as coordinates.