Continuous Rankin Bound for Hilbert and Banach Spaces

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} (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).

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