[Paper Review] Slanted matrices, Banach frames, and sampling

Akram Aldroubi, А. Г. Баскаков+1 more|ArXiv.org|May 29, 2007

Mathematical Analysis and Transform Methods21 references4 citations

TL;DR

This paper establishes that boundedness below of slanted matrices in ℓ^p spaces for any p ∈ [1, ∞] implies boundedness below for all p, enabling new results in Banach frames and irregular sampling. It proves a non-commutative Wiener's lemma for slanted matrices and derives explicit sampling bounds for functions generated by B-splines and other functions with controlled derivatives.

ABSTRACT

In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted matrices boundedness below of the corresponding operator in $\ell^p$ for some $p$ implies boundedness below in $\ell^p$ for all $p$. We use the established resultto enrich our understanding of Banach frames and obtain new results for irregular sampling problems. We also present a version of a non-commutative Wiener's lemma for slanted matrices.

Motivation & Objective

• To establish a general spectral property for slanted matrices: boundedness below in ℓ^p for one p implies boundedness below for all p.
• To apply this result to strengthen the theory of Banach frames and irregular sampling in function spaces.
• To derive explicit lower bounds for sampling operators in terms of sampling density and function regularity.
• To prove a non-commutative Wiener's lemma for slanted matrices, extending classical results to non-banded, structured infinite matrices.
• To provide quantitative conditions under which irregular sampling sets are uniformly stable across all ℓ^p and L^p spaces.

Proposed method

• Define slanted matrices as operator matrices (a_{mn}) with entries mapping between Banach spaces X_n and Y_m, indexed over ℤ^d.
• Use weighted ℓ^p norms and submultiplicative/Gelfand–Raikov–Shilov (GRS) weights to control growth and ensure boundedness.
• Establish operator norm estimates via Schur-type tests and spectral localization techniques for slanted matrices.
• Apply the theory to Paley–Wiener spaces and shift-invariant spaces generated by compactly supported functions.
• Use integral representations and fundamental theorems of calculus to bound pointwise values of functions in terms of coefficients and derivatives.
• Derive sampling bounds by estimating suprema of function values via L^∞ norms of derivatives and sampling mesh size γ(X).

Experimental results

Research questions

• RQ1 Does boundedness below of a slanted matrix in ℓ^p for a single p ∈ [1, ∞] imply boundedness below in all ℓ^p spaces?
• RQ2 Under what conditions is an irregular sampling set a set of sampling for all p ∈ {0} ∪ [1, ∞]?
• RQ3 Can a non-commutative Wiener's lemma be established for slanted matrices, generalizing classical results?
• RQ4 What explicit lower bounds can be derived for sampling operators in terms of the sampling mesh and function regularity?
• RQ5 How do the spectral properties of slanted matrices relate to the stability of reconstruction in Banach frames and shift-invariant systems?

Key findings

• Boundedness below of a slanted matrix in ℓ^p for any p ∈ [1, ∞] implies boundedness below in ℓ^p for all p, a key spectral invariance property.
• For a function φ ∈ W^1_ω with a‖c‖_∞ ≤ ‖∑c_k φ_k‖_∞ ≤ b‖c‖_∞ and ‖∑c_k φ''_k‖_∞ ≤ b''‖c‖_∞, sampling sets with γ(X) < 2a/b' yield lower sampling bounds in ℓ^∞.
• For the B-spline β₁, if γ(X) < 1, then X is a set of sampling for V^0(β₁) with lower bound 1 − γ(X), and this bound extends uniformly to all p ∈ [1, ∞].
• For the B-spline β₂, if γ(X) < 1, then X is a set of sampling for V^0(β₂) with lower bound ½(1 − γ²(X)), and this bound is uniform across all p ∈ [1, ∞].
• The paper proves a non-commutative Wiener’s lemma for slanted matrices, showing that invertibility in ℓ^p implies boundedness below uniformly across p.
• Explicit universal sampling bounds are derived for all p ∈ [1, ∞] using derivative estimates and mesh size γ(X), valid for functions in W^1_ω or W^2_ω.