I know it is possible to find a Fourier Series that defines a function though a set of 2D points (as shown here), but I want to know how to do this for a set of $n$ dimensional points.
I am assuming I would have to apply a Fourier transform on the data, but how is that information used to construct the function? In 2D space, the function is created by summing complex numbers that are computed using the coefficients.
$$f(x)\sim\frac{a_{0}}{2}+\sum_{k=1}^{\infty}\left(a_{k}\cos\left(kx\right)+b_{k}\sin\left(kx\right)\right)$$
What I need help with is generalizing that equation. I do not know how the nth dimension contributes to this function.
Thanks in advance!