Heisenberg's inequality

Question

The Heisenberg's inequality in $\Bbb{R}$ reads $$\|f\|_{L^2}^4\leq \int_{\Bbb{R}}x^2f(x)^2dx\int_{\Bbb{R}}\xi^2\hat{f}(\xi)^2d\xi$$ where by $\hat{f}$ we refer to the Fourier transform of $f$. The aformentioned inequality refered to as Heisenberg's since it is in consistency with the Heisenberg uncertainty principle which states that $$\sigma_x\sigma_\xi\geq \frac{\hbar}{2}$$ where $\hbar$ is the reduced Planck constant, $h/(2\pi)$). A simple interpretation of that is: $f$ and $\hat{f}$ can't both be concentrated in a small region. Now, consider the following which holds in periodic domain $\Bbb{T}$ $$\|f\|_{L^2(\Bbb{T})}^2=\sum_{k\in \mathbb{Z}}|c_k(f)|^2\leq \sup_{k\in \Bbb{Z}}|c_k(f)|\sum_{k\in \mathbb{Z}}|c_k(f)|\leq \underbrace{\|f\|_{L^1(\Bbb{T})}}_{space}\times \underbrace{\sum_{k\in \mathbb{Z}}|c_k(f)|}_{frequency}$$ does the aformentioned has anything to do with Heisenberg inequality and its interpretation since it involves as i indicated above the $L^2$ norm of $f$ controlled by a measurement in space multiplied by its counterpart in frequency space?