(Tao Vol.2, P.116, Q.5.3.1) Show that the product of any two trigonometric polynomials is again a trigonometric polynomial.
The author defines a trigonometric polynomial $f$ to be $f(x) : = \sum_{n= -N}^N c_n e_n$ where $c_n \in \mathbb{C}$ and $e_n(x) : = e^{2 \pi i n x}$. Suppose that $f$ and $g$ are trigonometric polynomials. Then,
$$f\cdot g = \sum_{n = -N}^Nc_ne_n \sum_{m=-M}^M d_me_m = \sum_{n = -N}^N \sum_{m=-M}^M c_nd_me_{n+m}.$$
I think that the coefficient should be some $z_{n+m}$ in $\mathbb{C}$ and it looks like a convolution of $c_n$ with $d_m$. But, I don't know how to get this. I would appreciate if you give some help.