English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces and $p=1$ or$p=\infty$. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in\Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in\Delta})$ be unbounded continuous p-Schauder frames for a Banach space$\mathcal{X}$. Then for every $x \in ( \mathcal{D}(\theta_f)\cap\mathcal{D}(\theta_g))\setminus\{0\}$, we show that \begin{align}\label{UB}  (1) \quad \quad \quad \quad \mu(\operatorname{supp}(\theta_fx))\nu(\operatorname{supp}(\theta_g x)) \geq\frac{1}{\left(\displaystyle\sup_{\alpha \in \Omega, \beta \in\Delta}|f_\alpha(\omega_\beta)|\right)\left(\displaystyle\sup_{\alpha \in\Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|\right)}, \end{align} where\begin{align*} &\theta_f:\mathcal{D}(\theta_f) \ni x \mapsto \theta_fx \in\mathcal{L}^p(\Omega, \mu); \quad \theta_fx: \Omega \ni \alpha \mapsto(\theta_fx) (\alpha):= f_\alpha (x) \in \mathbb{K},\\ &\theta_g:\mathcal{D}(\theta_g) \ni x \mapsto \theta_gx \in \mathcal{L}^p(\Delta, \nu);\quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in\mathbb{K}. \end{align*} We call Inequality (1) as \textbf{UnboundedDonoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Alongwith recent \textbf{Functional Continuous Uncertainty Principle}[arXiv:2308.00312], Inequality (1) also improves Ricaud-Torr\'{e}saniuncertainty principle [IEEE Trans. Inform. Theory, 2013]. In particular, itimproves Elad-Bruckstein uncertainty principle [IEEE Trans. Inform. Theory,2002] and Donoho-Stark uncertainty principle [SIAM J. Appl. Math., 1989].

Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'esani Uncertainty Principles