Discrete and Continuous Welch Bounds for Banach Spaces with Applications

Abstract

Let $\{\tau_j\}_{j=1}^n$ be a collection in a finite dimensional Banach space $\mathcal{X}$ of dimension $d$ and $\{f_j\}_{j=1}^n$ be a collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) such that $f_j(\tau_j) =1$, $\forall 1\leq j\leq n$. Let $n\geq d$ and $\text{Sym}^m(\mathcal{X})$ be the Banach space of symmetric m-tensors. If the operator $ \text{Sym}^m(\mathcal{X})\ni x \mapsto \sum_{j=1}^nf_j^{\otimes m}(x)\tau_j ^{\otimes m}\in\text{Sym}^m(\mathcal{X})$ is diagonalizable and its eigenvalues are all non negative, then we prove that \begin{align}\label{WELCHBANACHABSTRACT} \max _{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|^{2m}\geq \max _{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)f_k(\tau_j)|^m \geq\frac{1}{n-1}\left[\frac{n}{{d+m-1\choose m}}-1\right], \quad \forall m \in \mathbb{N}. \end{align} When $ \mathcal{X}=\mathcal{H}$ is a Hilbert space, and $f_j$ is defined by $f_j: \mathcal{H}\ni h \mapsto \langle h, \tau_j \rangle \in \mathbb{K}$ (where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$), $\forall 1 \leq j \leq n$, then Inequality (1) reduces to Welch bounds. Thus Inequality (1) improves 48 years old result obtained by Welch [\textit{IEEE Transactions on Information Theory, 1974}]. We also prove the following continuous version of Inequality (1) under certain conditions for measure spaces: \begin{align}\label{CONTINUOUSWELCHBANACHABSTRACT} \sup _{\alpha, \beta \in \Omega, \alpha\neq \beta}|f_\alpha(\tau_\beta) |^{2m}\geq \sup _{\alpha, \beta \in \Omega, \alpha\neq \beta}|f_\alpha(\tau_\beta)f_\beta(\tau_\alpha) |^{m}\geq \frac{1}{(\mu\times\mu)((\Omega\times\Omega)\setminus\Delta)}\left[\frac{ \mu(\Omega)^2}{{d+m-1 \choose m}}-(\mu\times\mu)(\Delta)\right]. \end{align}