Discrete and Continuous Welch Bounds for Banach Spaces with Applications

TL;DR

This paper generalizes Welch bounds from Hilbert spaces to Banach spaces, providing discrete and continuous inequalities that improve upon the classical 1974 result and have applications in frame theory and signal processing.

Contribution

It introduces new Welch-type bounds for Banach spaces, extending classical results and establishing continuous versions under measure-theoretic conditions.

Findings

Derived discrete Welch bounds for Banach spaces.

Established continuous Welch bounds with measure space conditions.

Improved classical Welch bounds by a factor depending on space dimension and tensor order.

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}