Let $(\{ f_j\}_{ j=1}^n, \{ \tau_j\}_{ j=1}^n)$ and $(\{ g_k\}_{ k=1}^n, \{ \omega_k\}_{ k=1}^n)$ be two p-orthonormal bases for a finite dimensional Banach space $\mathcal{ X}$. Let $M,N\subseteq \{ 1, \dots, n\}$ be such that \begin{ align*} o(M)^\frac{ 1}{ q}o(N)^\frac{ 1}{ p}< \frac{ 1}{ \displaystyle \max_{ 1\leq j,k\leq n}|g_k(\tau_j) |}, \end{ align*} where $q$ is the conjugate index of $p$. Then for all $x \in \mathcal{ X}$, we show that \begin{ align}\label{ FGJU} (1) \quad \quad \quad \quad \|x\|\leq \left(1+\frac{ 1}{ 1-o(M)^\frac{ 1}{ q}o(N)^\frac{ 1}{ p}\displaystyle\max_{ 1\leq j,k\leq n}|g_k(\tau_j)|}\right)\left[\left(\sum_{ j\in M^c}|f_j(x)|^p\right)^\frac{ 1}{ p}+\left(\sum_{ k\in N^c}|g_k(x) |^p\right)^\frac{ 1}{ p}\right]. \end{ align} We call Inequality (1) as \textbf{ Functional Ghobber-Jaming Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Ghobber and Jaming \textit{ [Linear Algebra Appl., 2011]}.
Categories: math.FA cs.IT math.IT