This is related to a previously answered question here answered by @WillieWong

Edit 2: To focus the question, assume that the function maps into a finite subset of the unit circle. Let us simply take a $\pm 1$ valued function.

Edit 1 (June 2019): To avoid obvious divisibility issues as in the counterexample by @fedja in the comments let us assume $N$ is not divisible by any integer in $[1,m]$. So, let us take $N$ to be one less than the corresponding primorial, $N=m\#-1.$

Let $f:\mathbb{Z}\rightarrow \mathbb{C}.$ Assume that the support of $f$ is finite, say it is contained in $[1,N],$ it can even be taken to be $[1,N]$ if it helps, and that $\mid f\mid$ is not only nonzero but essentially constant on its support.

Define the fourier transform $$\widehat{f}:[0,1)\rightarrow \mathbb{C}$$ by $$\widehat{f(t)}=\sum_{n\in \mathbb{Z}} f(n)~e^{2i \pi n t}.$$

Now let $v$ be a positive integer $\geq 2,$ and let the "projected" function be $$ f_v(n)=\left\{ \begin{array}{ccc} f(n), & \quad\mathrm{if}\quad & v|n,\\ & & \\ 0 & & \mathrm{otherwise}. \end{array} \right. $$ Write $f=f_1$ for notational simplicity.

I am interested in a specific Parseval type relationship for this function, maybe expressed in terms of the transform of the original function?

Specifically, can one obtain a nontrivial bound of the form $$ \sum_{v=1}^m \mid \sum_{n \in \mathbb{Z}} f_v(n) \mid^2 {\geq} A(N,m) \int_0^1 \mid\widehat{f(t')} \mid^2 \,dt' $$ using some kind of uncertainty relation.

We can take $m\ll N,$ a fractional power of $N$ or even a power of $\log N.$

This is a followup to this earlier question

Let $f:\mathbb{Z}\rightarrow \{\pm 1\}.$ Assume that the support of $f$ is finite, say it is contained in $[1,N],$ it can even be taken to be $[1,N]$ if it helps. To avoid divisibility issues, assume that $N$ is is not divisible by any integer in $[1,m]$. One could, for example, take $N=m\#-1,$ where $m\#$ is the $m^{th}$ primorial.

Define the fourier transform $\widehat{f}$ on $[0,1)$ by $\widehat{f(t)}=\sum_{n\in \mathbb{Z}} f(n)~e^{-2i \pi n t}.$

Now let $v$ be a positive integer $\geq 2,$ and let the "projected" function be $$ f_v(n)=\left\{ \begin{array}{ccc} f(n), & \quad\mathrm{if}\quad & v|n,\\ & & \\ 0 & & \mathrm{otherwise}. \end{array} \right. $$ Of course $f_1$ is simply $f.$ Can one obtain a nontrivial bound of the form $$ \sum_{v=1}^m \mid \sum_{n \in \mathbb{Z}} f_v(n) \mid^2 {\geq} A(N,m) \int_0^1 \mid\widehat{f(t')} \mid^2 \,dt' $$ or similar, using some kind of uncertainty relation.

We can take $m\ll N,$ a fractional power of $N$ or even a power of $\log N.$

This is related to a previously answered question here answered by @WillieWong

Let $f:\mathbb{Z}\rightarrow \mathbb{C}.$ Assume that the support of $f$ is finite, say it is contained in $[1,N],$ it can even be taken to be $[1,N]$ if it helps, and that $\mid f\mid$ is not only nonzero but essentially constant on its support.

Define the fourier transform $$\widehat{f}:[0,1)\rightarrow \mathbb{C}$$ by $$\widehat{f(t)}=\sum_{n\in \mathbb{Z}} f(n)~e^{2i \pi n t}.$$

Now let $v$ be a positive integer $\geq 2,$ and let the "projected" function be $$ f_v(n)=\left\{ \begin{array}{ccc} f(n), & \quad\mathrm{if}\quad & v|n,\\ & & \\ 0 & & \mathrm{otherwise}. \end{array} \right. $$ Write $f=f_1$ for notational simplicity.

I am interested in a specific Parseval type relationship for this function, maybe expressed in terms of the transform of the original function?

Specifically, can one obtain a nontrivial bound of the form $$ \sum_{v=1}^m \mid \sum_{n \in \mathbb{Z}} f_v(n) \mid^2 {\geq} A(N,m) \int_0^1 \mid\widehat{f(t')} \mid^2 \,dt' $$ using some kind of uncertainty relation.

We can take $m\ll N,$ a fractional power of $N$ or even a power of $\log N.$

Edit: To avoid obvious divisibility issues as in the counterexample by @fedja in the comments let us assume $N$ is not divisible by any integer in $[1,m]$. We can take $N$ to be one less than the corresponding primorial, $N=m\#-1.$

This is related to a previously answered question here answered by @WillieWong

Edit 2: To focus the question, assume that the function maps into a finite subset of the unit circle. Let us simply take a $\pm 1$ valued function.

Edit 1 (June 2019): To avoid obvious divisibility issues as in the counterexample by @fedja in the comments let us assume $N$ is not divisible by any integer in $[1,m]$. So, let us take $N$ to be one less than the corresponding primorial, $N=m\#-1.$

Let $f:\mathbb{Z}\rightarrow \mathbb{C}.$ Assume that the support of $f$ is finite, say it is contained in $[1,N],$ it can even be taken to be $[1,N]$ if it helps, and that $\mid f\mid$ is not only nonzero but essentially constant on its support.

Define the fourier transform $$\widehat{f}:[0,1)\rightarrow \mathbb{C}$$ by $$\widehat{f(t)}=\sum_{n\in \mathbb{Z}} f(n)~e^{2i \pi n t}.$$

Now let $v$ be a positive integer $\geq 2,$ and let the "projected" function be $$ f_v(n)=\left\{ \begin{array}{ccc} f(n), & \quad\mathrm{if}\quad & v|n,\\ & & \\ 0 & & \mathrm{otherwise}. \end{array} \right. $$ Write $f=f_1$ for notational simplicity.

I am interested in a specific Parseval type relationship for this function, maybe expressed in terms of the transform of the original function?

Specifically, can one obtain a nontrivial bound of the form $$ \sum_{v=1}^m \mid \sum_{n \in \mathbb{Z}} f_v(n) \mid^2 {\geq} A(N,m) \int_0^1 \mid\widehat{f(t')} \mid^2 \,dt' $$ using some kind of uncertainty relation.

We can take $m\ll N,$ a fractional power of $N$ or even a power of $\log N.$