# The frame of truncated momentum basis on a 1D lattice

$$\def\ket#1{\left|#1\right\rangle } \def\bra#1{\left\langle #1\right|}$$ (This is part of a research problem)

The Setup: Consider a single particle on a finite 1D lattice with the Hilbert space spanned by the orthonormal position basis $$\ket{n;x}$$ for $$n=0,1,...,D-1$$. We define the lattice momentum basis by applying the lattice Fourier transform $$\ket{m;p}:=F\ket{m;x}:=\frac{1}{\sqrt{D}}\sum_{n=0}^{D-1}e^{i2\pi mn/D}\ket{n;x}.$$ We can then subdivide the lattice (in position space) into equidistant intervals of length $$\omega$$ and restrict the momentum basis to these position-space intervals, resulting in the definition of truncated momentum states $$\ket{m,k;p}:=\frac{1}{\sqrt{\omega}}\sum_{n\in\Omega_{k}}e^{i2\pi mn/D}\ket{n;x}$$ where $$\Omega_{k}:=\left\{ \omega k,\omega k+1,..., \omega k+\omega-1 \right\}$$ are the position indices of the $$k$$th interval.

We now focus on the subspace of a single interval spanned by $$\left\{\ket{n;x}\right\}_{n\in\Omega_{0}}$$. The truncated momentum basis are not pairwise orthogonal on this subspace as we can calculate $$\bra{m',0;p}\ket{m,0;p} =\frac{1}{\omega}\sum_{n\in\Omega_{0}}e^{i2\pi n\left(m-m'\right)/D}=\frac{1}{\omega}\frac{e^{i2\pi\left(m-m'\right)\omega/D}-1}{e^{i2\pi\left(m-m'\right)/D}-1}.$$

The Problem: From a combination of analysis and numerics I have reasons to believe that for $$\omega>>1$$ the following frame operator $$S$$ converges to a (unnormalized) projection $$S:=\sum_{m\in\Omega_{0}}\ket{m,0;p}\bra{m,0;p}\approx \frac{D}{\omega} \sum_{m=0...\gamma-1}\ket{m;p_{0}}\bra{m;p_{0}}$$ where $$\gamma=\frac{\omega^2}{D}$$ and I have introduced the native (orthonormal) momentum basis of the interval $$\ket{m;p_{0}}:=\frac{1}{\sqrt{\omega}}\sum_{n\in\Omega_{0}}e^{i2\pi mn/\omega}\ket{n;x}.$$ What would be a good strategy to argue / prove that the frame operator $$S$$ indeed converges to that projection?

Remarks:

1. I am not showing my reasons to believe in the convergence because it is messy and long. The problem is about finding an effective line of reasoning.
2. I believe that this problem should be known in the context of frame theory and signal processing. Any relevant results in that area are welcome.
3. Since this is a research problem, I'm not expecting a solution. Any thoughts on how would one approach this problem will be greatly appreciated (but who knows, I won't be too surprised to see a solution from people in this community).