Assume there are $N$ series $X_{1},X_{2},X_{3}...X_{N}$ (Each series would be of the form $X_{k} = [(t_{1},p_{1}),(t_{2},p_{2})...,(t_{m},p_{m})]$ .Series could be non-uniformly spaced) and their corresponding fourier transforms be $F_{X_{1}},F_{X_{2}},F_{X_{3}}...F_{X_{N}}$ . Can we find a relationship between the fourier transform of the combination of the series (ie. $F_{[X_{1};X_{2};X_{3}....;X_{N}]}$) and their individual fourier transforms (ie. $F_{X_{1}},F_{X_{2}},F_{X_{3}}...F_{X_{N}}$)?
$X_{1};X_{2};X_{3}....;X_{N}$ = union of all elements in $X_{1},X_{2},X_{3}...X_{N}$ sorted on t
If it is not possible to establish a relationship for the generic case, is it possible to do that after we impose any constraints on the series?