Continuous Rankin Bound for Hilbert and Banach Spaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866918031040446464 |
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| author | Krishna, K. Mahesh |
| author_facet | Krishna, K. Mahesh |
| contents | Let $(Ω, μ)$ be a measure space and $\{τ_α\}_{α\in Ω}$ be a normalized continuous Bessel family for a real Hilbert space $\mathcal{H}$. If the diagonal $Δ:= \{(α, α):α\in Ω\}$ is measurable in the measure space $Ω\times Ω$, then we show that \begin{align} (1) \quad\quad\quad\quad \sup _{α, β\in Ω, α\neq β}\langle τ_α, τ_β\rangle \geq \frac{-(μ\timesμ)(Δ)}{(μ\timesμ)((Ω\timesΩ)\setminusΔ)}. \end{align} We call Inequality (1) as continuous Rankin bound. It improves 76 years old result of Rankin [\textit{Ann. of Math., 1947}]. It also answers one of the questions asked by K. M. Krishna in the paper [Continuous Welch bounds with applications, \textit{Commun. Korean Math. Soc., 2023}]. We also derive Banach space version of Inequality (1). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07606 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuous Rankin Bound for Hilbert and Banach Spaces Krishna, K. Mahesh Functional Analysis 42C15 Let $(Ω, μ)$ be a measure space and $\{τ_α\}_{α\in Ω}$ be a normalized continuous Bessel family for a real Hilbert space $\mathcal{H}$. If the diagonal $Δ:= \{(α, α):α\in Ω\}$ is measurable in the measure space $Ω\times Ω$, then we show that \begin{align} (1) \quad\quad\quad\quad \sup _{α, β\in Ω, α\neq β}\langle τ_α, τ_β\rangle \geq \frac{-(μ\timesμ)(Δ)}{(μ\timesμ)((Ω\timesΩ)\setminusΔ)}. \end{align} We call Inequality (1) as continuous Rankin bound. It improves 76 years old result of Rankin [\textit{Ann. of Math., 1947}]. It also answers one of the questions asked by K. M. Krishna in the paper [Continuous Welch bounds with applications, \textit{Commun. Korean Math. Soc., 2023}]. We also derive Banach space version of Inequality (1). |
| title | Continuous Rankin Bound for Hilbert and Banach Spaces |
| topic | Functional Analysis 42C15 |
| url | https://arxiv.org/abs/2311.07606 |