Let $f, g$ be continuous $1$-periodic functions. Show that $$\widehat{fg}(n)=\sum_{m\in \mathbb{Z}}\hat{f}(n-m)\hat{g}(m).$$
Here the $\widehat{f\cdot g}$ denotes the Fourier coefficient of $f\cdot g$, and $\hat{f}$ and $\hat{g}$ the Fourier coefficients of $f$ and $g$, respectively.
I am able to show that the sum converges (via Cauchy-Schwarz). But I am having trouble showing equality. Does anyone have a hint to point me in the right direction?