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} (1) \quad\quad\quad\quad \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:= \{ z\in \mathcal{ X}: f_j(z)\neq 0, 1\leq j \leq n\}, \quad \mathcal{ X}_g:= \{ w\in \mathcal{ X}: g_k(w)\neq 0, 1\leq k \leq m\},\\ &S_f (x):= -\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):= -\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 (1) as \textbf{ Functional Deutsch Uncertainty Principle}. For Hilbert spaces, we show that Inequality (1) reduces to the uncertainty principle obtained by Deutsch \textit{ [Phys. Rev. Lett., 1983]}. We also derive a dual of Inequality (1).
Categories: math.FA