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 $D_{n} = \lbrace j \in \mathbb{Z} : 2 \pi j/ n \in (- \pi, \pi] \rbrace$.

  • $\begingroup$ See how the terms $e^{-i(t_1-t_2) \lambda}$ appear in $F_x(\lambda)F_y(u-\lambda)$ then for $0 \ne t_1-t_2 $ dividing $ n!$ : $\sum_{k= 1}^{n!} e^{-i(t_1-t_2) 2\pi k/n!} =0$. You can replace $n!$ by $lcm(1,\ldots,n)$ but not by $n$. $\endgroup$ – reuns Dec 9 '18 at 1:17
  • $\begingroup$ I tried to go from either ways to the other, but miserabely failed. Would you be able to help more ? $\endgroup$ – mich95 Dec 9 '18 at 1:20
  • $\begingroup$ Where are you stuck in what I said ? $\endgroup$ – reuns Dec 9 '18 at 1:21
  • $\begingroup$ I do not get any $n!$... $\endgroup$ – mich95 Dec 9 '18 at 2:19

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