If $a=(a_{1}, \ldots,a_{n})$. Define $F_{a}( \lambda)= n^{-1/2} \sum\limits_{t=1}^{n} a_{t}e^{-it \lambda}$.
Let $\lbrace x_{1}, \ldots,x_{n} \rbrace$ and $\lbrace y_{1},\ldots,y_{n}\rbrace$ be real numbers. Let $z_{t}=x_{t}y_{t}$. I want to show the following equality, $F_{z}(2 \pi j /n)= n^{-1/2} \sum\limits_{ k \in D_{n}} F_{X}( 2 \pi j/n)F_{Y}(2 \pi (j-k)/n)$, where $D_{n} = \lbrace j \in \mathbb{Z} : 2 \pi j/ n \in (- \pi, \pi] \rbrace$.