Functional Ghobber–Jaming Uncertainty Principle

Let \(\bigl(\{f_j\}_{j=1}^n, \{\tau_j\}_{j=1}^n\bigr)\) and \(\bigl(\{g_k\}_{k=1}^n, \{\omega_k\}_{k=1}^n\bigr)\) be two \(p\) -orthonormal bases for a finite-dimensional Banach space \(\mathcal{X}\) . Let \(M,N\subseteq \{1, \ldots, n\}\) be such that \(o(M)^{\tfrac{1}{q}}o(N)^{\tfrac{1}{p}}< \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|g_k(\tau_j) |},\) where \(q\) is the conjugate index of \(p\) and \(o(M)\) is the cardinality of \(M\) . Then for all \(x \in \mathcal{X}\) we show that \(\|x\|\leq \Biggl(1+\frac{1}{1-o(M)^{\tfrac{1}{q}}o(N)^{\tfrac{1}{p}} \displaystyle\max_{1\leq j,k\leq n}|g_k(\tau_j)|}\Biggr)\left[\Biggl(\sum_{j\in M^c}|f_j(x)|^p\Biggr)^{\tfrac{1}{p}}+\Biggl(\sum_{k\in N^c}|g_k(x) |^p\Biggr)^{\tfrac{1}{p}}\right].\) We refer to this inequality as the Functional Ghobber–Jaming Uncertainty Principle. This inequality improves the uncertainty principle obtained by Ghobber and Jaming [Linear Algebra Appl., 2011].