I try to see how
\begin{align*} 3^{-n} \sum_{Y \in \mathbb{Z}/3^n \mathbb{Z}}|g(Y)-3^{m-n}\sum_{Y' \in \mathbb{Z}/3^n \mathbb{Z} |Y' = Y \mod 3^m} g(Y')|^2 = \\ = \sum_{\xi \in \mathbb{Z}/3^n \mathbb{Z} : 3^{n-m} \nmid \xi}|\sum_{Y \in \mathbb{Z}/3^n \mathbb{Z}} g(Y)e^{-2 \pi i \xi Y / 3^n} |^2 \end{align*}
follows from Plancherel's theorem, which at least I know only in the form \begin{align*} \int_{-\infty}^{\infty} f(x) \bar{g(x)} dx= \int_{-\infty}^{\infty} \hat{f(\xi)} \bar{\hat{g(\xi)}} d \xi. \end{align*} Having only little experience in harmonic analysis, I would appreciate any help.