Multi-Variable / Dimension Fourier Transform

Say we have f(x, y). We can Fourier decompose it in terms of f1(y, v) and $e^{\ x\ v}$, f2(x, u) and $e^{\ u\ y}$, or both variables simultaneously f3(u, v) and $e^{\ x\ v\ +\ u\ y}$. Similarly for any greater number of variables or dimensions. Now, is there any proof or derivation for all of this (first of all existence, but uniqueness would be nice too)? I've always wondered why the decomposition exactly matches up to the original function, especially when decomposing > 1 variables together. Does it matter the order of the variables with which we decompose / integrate? There must be some sort of mechanism, for the lack of a better term, making sure that the individual 1-D decompositions interfere with each other exactly to yield the original function, no?

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 Anyone knows? ... Btw I left out the imaginary number in the OP; corrected below. Say we have f(x, y). We can Fourier decompose it in terms of f1(y, v) and $e^{\ i\ x\ v}$, f2(x, u) and $e^{\ i\ u\ y}$, or both variables simultaneously f3(u, v) and $e^{\ i\ (x\ v\ +\ u\ y)}$. Similarly for any greater number of variables or dimensions. Now, is there any proof or derivation for all of this (first of all existence, but uniqueness would be nice too)? I've always wondered why the decomposition exactly matches up to the original function, especially when decomposing > 1 variables together. Does it matter the order of the variables with which we decompose / integrate? There must be some sort of mechanism, for the lack of a better term, making sure that the individual 1-D decompositions interfere with each other exactly to yield the original function, no?
 Check out fubini's theorem about switching integration order

 Tags dimension, fourier, multi, variable