# Parseval Relation for Dilated Function on Integers

Let $f:\mathbb{Z}\rightarrow \mathbb{C}.$ If necessary assume that the support of $f$ is finite, and that $\mid f\mid$ is bounded. Define the fourier transform $$\hat{f}:[0,1)\rightarrow \mathbb{C}$$ by $$\hat{f}=\sum_{n\in \mathbb{Z}} f(n)~e^{2i \pi n t}.$$

Parseval states that $$\sum_{n \in \mathbb{Z}} \mid f(n) \mid^2 = \int_0^1 \mid\hat{f}(t) \mid^2 \,dt$$ holds. Now let $v$ be a positive integer $\geq 2,$ and let $$f_v(n)=\left\{ \begin{array}{ccc} f(n/v), & \quad\mathrm{if}\quad & v|n,\\ & & \\ 0 & & \mathrm{otherwise}. \end{array} \right.$$ What is the Parseval relationship for this function?

Also define, with $v$ as before, the "sampled" function $$g_v(n)=\left\{ \begin{array}{ccc} f(n), & \quad\mathrm{if}\quad & v|n,\\ & & \\ 0 & & \mathrm{otherwise}. \end{array} \right.$$ What is the Parseval relationship for this function?

Let $$f_v(n)=\left\{ \begin{array}{ccc} f(n), & \quad\mathrm{if}\quad & v|n,\\ & & \\ 0 & & \mathrm{otherwise}. \end{array} \right.$$ Let $$f=f_1$$ for simplicity's sake. Then
$$\sum_{v=1}^m \sum_{n \in \mathbb{Z}} \mid f_v(n) \mid^2 \geq \left( \ln m-\frac{m}{N}\right) \int_0^1 \mid\widehat{f(t)} \mid^2 \,dt.$$