Functional Ghobber-Jaming Uncertainty Principle

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(τ_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(τ_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 $\mathbf{\text{Functional}}\mathbf{\text{Ghobber-Jaming}}\mathbf{\text{Uncertainty}}\mathbf{\text{Principle}}$. Inequality (1) improves the uncertainty principle obtained by Ghobber and Jaming $\mathit{\text{[Linear}}\mathit{\text{Algebra}}\mathit{\text{Appl.,}}\mathit{\text{2011]}}$.