I have a vector field ${\bf{f}}(x,y)=(f_x(x,y),f_y(x,y))$, I would like to express it in terms of its Fourier coefficients, $f_x(x,y) = \sum_n h_x(n) g_n(x,y)$, $f_y(x,y) = \sum_n h_y(n) g_n(x,y)$ where $g_n$ is the $n$th basis. I take ${\bf{h}}_n=(h_x(n),h_y(n))$.
Now I want the correct notation to write something like ${\bf f}= {\bf h}\otimes {\bf g}$, where some form of product should replace $\otimes$, and ${\bf h}=\oplus_n{\bf h}_n$, ${\bf g}=\oplus_n{\bf g}_n$ refer to vector of vector valued coefficients and vector of basis functions respectively.
All I have written down are probably incorrect. I hope they can be corrected and clarified.