2022/01/04 by K. Mahesh Krishna, Krishna, K. Mahesh
Engineering · Mathematics · Medicine · #42C15 #Advanced Neuroimaging Techniques and Applications #Elasticity and Material Modeling #FOS: Mathematics #Functional Analysis (math.FA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2201.00980
openalex publication_date 2022/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \τj\j=1n be a collection in a finite dimensional Banach space X of dimension d and \fj\j=1n be a collection in X^* (dual of X) such that fj(τj) =1, ∀ 1≤ j≤ n. Let n≥ d and Symm(X) be the Banach space of symmetric m-tensors. If the operator Symm(X)\ni x ↦ ∑j=1nfj⊗ m(x)τj ⊗ m\inSymm(X) is diagonalizable and its eigenvalues are all non negative, then we prove that max 1≤ j,k ≤ n, j≠ k|fj(τk)|2m≥ max 1≤ j,k ≤ n, j≠ k|fj(τk)fk(τj)|m ≥(1)/(n-1)[(n)/(d+m-1\choose m)-1], ∀ m ∈ ℕ. When X=H is a Hilbert space, and fj is defined by fj: H\ni h ↦ ⟨ h, τj ⟩ ∈ \mathbbK (where \mathbbK is ℝ or ℂ), ∀ 1 ≤ j ≤ n, then Inequality (1) reduces to Welch bounds. Thus Inequality (1) improves 48 years old result obtained by Welch [IEEE Transactions on Information Theory, 1974]. We also prove the following continuous version of Inequality (1) under certain conditions for measure spaces: sup α, β∈ Ω, α≠ β|fα(τβ) |2m≥ sup α, β∈ Ω, α≠ β|fα(τβ)fβ(τα) |m≥ (1)/((μ×μ)((Ω×Ω)∖Δ))[( μ(Ω)2)/(d+m-1 \choose m)-(μ×μ)(Δ)].