Article
Version 1
Preserved in Portico This version is not peer-reviewed
Continuous Rankin Bound for Hilbert and Banach Spaces
Version 1
: Received: 20 August 2023 / Approved: 21 August 2023 / Online: 21 August 2023 (07:12:22 CEST)
How to cite: KRISHNA, K.M. Continuous Rankin Bound for Hilbert and Banach Spaces. Preprints 2023, 2023081426. https://doi.org/10.20944/preprints202308.1426.v1 KRISHNA, K.M. Continuous Rankin Bound for Hilbert and Banach Spaces. Preprints 2023, 2023081426. https://doi.org/10.20944/preprints202308.1426.v1
Abstract
Let $(Omega, mu)$ be a measure space and $\{\tau_\alpha\}_{\alpha\in \Omega}$ be a normalized continuous Bessel family for a real Hilbert space $\mathcal{H}$. If the diagonal $\Delta \{(\alpha, \alpha):\alpha \in \Omega\}$ is measurable in the measure space $\Omega\times \Omega$, then we show that \begin{align}\label{CRBA} \sup _{\alpha, \beta \in \Omega, \alpha\neq \beta}\langle \tau_\alpha, \tau_\beta\rangle \geq \frac{-(\mu\times\mu)(\Delta)}{(\mu\times\mu)((\Omega\times\Omega)\setminus\Delta)}. \end{align} We call Inequality (1) as continuous Rankin bound. It improves 76 years old result of Rankin [Ann. of Math., 1947]. It also answers one of the questions asked by K. M. Krishna in the paper [Continuous Welch bounds with applications, Commun. Korean Math. Soc., 2023]. We also derive Banach space version of Inequality (1).
Keywords
Rankin bound; Continuous Bessel family
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Comments (0)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
* All users must log in before leaving a comment
