A multivariate Fourier Series of $f(\mathbf{u})$ can be defined by:
$$f(\mathbf{u}) = \sum_{n\in \mathbb{Z}^{d}}^\infty C_n e^{2\pi i n \mathbf{u}}$$
where:
$$f: \mathbb{R}^{d} \to \mathbb{R}$$
My question is: how to use Fourier Series to approximate a function such as
$$f: \mathbb{R}^{d} \to \mathbb{R}^{q}$$
where $q > 1$?
For instance, is there a way to use Fourier Series to approximate a function like this:
$$f(x,y,z)=(x^3+zy^3, x^2z + y)$$
?
Is it safe solving it by doing:
$$f(x,y,z)=(g(x,y,z), h(x,y,z))$$
$$g(x,y,z) = \sum_{n,m,o}C_g^{mno} e^{2\pi i (mx+ny+oz) }$$
$$h(x,y,z) = \sum_{n,m,o}C_h^{mno} e^{2\pi i (mx+ny+oz) }$$
?