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 ${\mathcal{X}}^{\ast }$ (dual of $\mathcal{X}$) such that $f_j({\tau }_j)=1$, $\forall 1\le j\le n$. Let $n\ge d$ and ${\text{Sym}}^m(\mathcal{X})$ be the Banach space of symmetric m-tensors. If the operator ${\text{Sym}}^m(\mathcal{X})\ni x\mapsto {\sum }_{j=1}^n{f}_j^{\otimes m}(x){\tau }_j^{\otimes m}\in {\text{Sym}}^m(\mathcal{X})$ 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(τ_k)|^{2m}\geq \max _{1\leq j,k \leq n, j\neq k}|f_j(τ_k)f_k(τ_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 $f_j:\mathcal{H}\ni h\mapsto ⟨h,{\tau }_j⟩\in \mathbb{K}$ (where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$), $\forall 1\le j\le n$, 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 _{α, β\in Ω, α\neq β}|f_α(τ_β) |^{2m}\geq \sup _{α, β\in Ω, α\neq β}|f_α(τ_β)f_β(τ_α) |^{m}\geq \frac{1}{(μ\timesμ)((Ω\timesΩ)\setminusΔ)}\left[\frac{ μ(Ω)^2}{d+m-1 \choose m}-(μ\timesμ)(Δ)\right]. \end{align}

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References in corpus (3)

• Modular Welch Bounds with Applications
• Construction of Continuous Frames in Hilbert spaces
• Unconditional structures of translates for $L_p(R^d)$

Cited by in corpus (1)

• p-adic Welch Bounds and p-adic Zauner Conjecture