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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.
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