preprints.org > computer science and mathematics > analysis > doi: 10.20944/preprints202308.1108.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 12 August 2023 / Approved: 14 August 2023 / Online: 15 August 2023 (09:08:33 CEST)

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be continuous 1-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align*} \mu(\operatorname{supp}(\theta_f x)) \geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|}, \quad \nu(\operatorname{supp}(\theta_g x))\geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|}. \end{align*} where \begin{align*} &\theta_f: \mathcal{X} \ni x \mapsto \theta_fx \in \mathcal{L}^1(\Omega, \mu); \quad \theta_fx: \Omega \ni \alpha \mapsto (\theta_fx) (\alpha):= f_\alpha (x) \in \mathbb{K},\\ &\theta_g: \mathcal{X} \ni x \mapsto \theta_gx \in \mathcal{L}^1(\Delta, \nu); \quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K}. \end{align*} This solves a problem asked by K. M. Krishna in the paper 'Functional Continuous Uncertainty Principle' \textit{[arXiv:2308.00312v1]}.

Uncertainty Principle, Parseval Frame, Banach space.

Computer Science and Mathematics, Analysis

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