Continuous Rankin Bound for Hilbert and Banach Spaces
K. Mahesh Krishna
TL;DR
This paper generalizes Rankin's unit-vector bound to continuous settings by introducing a continuous Rankin bound for normalized continuous Bessel families indexed by a measure space, under the condition that the diagonal is measurable. It proves a Hilbert-space version using continuous frames, analysis and synthesis operators, and Fubini's theorem, yielding a dimension-free bound on cross-inner products $\sup_{\alpha\neq \beta} \langle \tau_\alpha, \tau_\beta\rangle$ and a corresponding bound on pairwise distances $\inf_{\alpha\neq \beta} \|\tau_\alpha-\tau_\beta\|^2$. The work also extends the bound to Banach spaces via a functional, continuous $p$-approximate Bessel framework, providing a bound on cross-action $f_\alpha(\tau_\beta)$ and a finite-$n$ analogue $\max_{j\neq k} f_j(\tau_k) \ge -1/(n-1)$. Together, these results connect continuous Welch-type bounds with Rankin-type limits and address questions raised by Krishna, broadening the applicability to both Hilbert and Banach space settings.
Abstract
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).