There is a famous low pass filter $[1,2,1]$ in signal processing which can be factored in the sense of a convolution product over the real numbers : $[1,1] * [1,1]$. This is the only way to do it over the reals ( I think ). Today when I was bored, I investigated some complex valued filters and came to the conclusion that for $k = 3$ and $k = 5$ and $z_0 = e^{2\pi i/k}$, i.e. the "first" of the roots of unity for degree 3 and 5, I can build the same filter like:
$$[1,2,1] = \left(z_0^{[m,l]} * z_0^{[-m,-l]}\right) \text{ where } m-l = (0 \text{ mod } k)$$ with the notation $$z^{[a,b]} = [z^a,z^b]$$ What is cool is that there is a certain symmetry between these filters.
Now my question is two-fold :
1) Can we prove this will work in general for any $k \in \mathbb{N}$? ( I suspect it is true ).
2) Do there exist any techniques to factor other filters in any systematic fashion? Do we need to expand the complex numbers to do so or will they suffice?
Edit The answer is quite trivial as pointed out by MattL, but maybe can give raise to more interesting questions or something. Here is a more interesting question (I hope).