Abstract:
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
\begin{equation}
\tag{1}
\|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].
\end{equation}
We refer to inequality (1) as the Functional Ghobber–Jaming
Uncertainty Principle.
Inequality (1) improves the uncertainty
principle obtained by Ghobber and Jaming [Linear Algebra Appl., 2011].
