FUNCTIONAL DEUTSCH UNCERTAINTY PRINCIPLE

Let $\{f_j\}_{j=1}^n$ and $\{g_k\}_{k=1}^m$ be  Parseval p-frames  for a finite dimensional Banach space $\mathcal{X}$. Then   we show that 
\begin{align}\label{UE}
 \log (nm)\geq S_f (x)+S_g (x)\geq -p \log \left(\displaystyle\sup_{y \in \mathcal{X}_f\cap \mathcal{X}_g, \|y\|=1}\left(\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(y)g_k(y)|\right)\right), \quad \forall x \in \mathcal{X}_f\cap \mathcal{X}_g,
\end{align}
where 
\begin{align*}
    &\mathcal{X}_f\coloneqq \{z\in \mathcal{X}: f_j(z)\neq 0, 1\leq j \leq n\}, \quad  \mathcal{X}_g\coloneqq \{w\in \mathcal{X}: g_k(w)\neq 0, 1\leq k \leq m\},\\
&S_f (x)\coloneqq -\sum_{j=1}^{n}\left|f_j\left(\frac{x}{\|x\|}\right)\right|^p\log \left|f_j\left(\frac{x}{\|x\|}\right)\right|^p, \quad   S_g (x)\coloneqq -\sum_{k=1}^{m}\left|g_k\left(\frac{x}{\|x\|}\right)\right|^p\log \left|g_k\left(\frac{x}{\|x\|}\right)\right|^p, \quad  \forall x \in \mathcal{X}_g.
\end{align*}
 We call Inequality (\ref{UE}) as \textbf{Functional Deutsch Uncertainty Principle}. For Hilbert spaces, we show that Inequality (\ref{UE})  reduces to the uncertainty principle  obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We also derive a dual of Inequality  (\ref{UE}).