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Variations on two Cabrelli's works

Elona Agora, Jorge Antezana, Diana Carbajal

TL;DR

The paper advances the theory of shift-invariant spaces by (i) deriving a triangular (and in the normal case, diagonal) decomposition for shift-preserving operators on finitely generated SIS via a fibered range-operator framework, and (ii) providing a practical ambient-space criterion in $\mathbb{T}^{k\times k}$ for when a multi-tiling set $\Omega$ yields a structured Riesz basis of exponentials with periodic frequencies in $L^2(\Omega)$. By reducing global questions to pointwise finite-dimensional problems on fibers, the authors unify operator canonical forms with multi-tiling spectral analysis and Paley-Wiener space structure, enabling explicit constructions and necessary/sufficient conditions grounded in determinant tests and a simple geometric separation property. The results generalize and sharpen previous decompositions (e.g., ACCP) and offer a tangible criterion for structured exponential bases in sampling and tiling contexts, with extensions to LCA groups. Overall, the work bridges shift-invariant theory, tiling, and Paley-Wiener analysis to yield actionable decompositions and basis characterizations for SIS and their spectral sets.

Abstract

In this paper we present two different problems within the framework of shift-invariant theory. First, we develop a triangular form for shift-preserving operators acting on finitely generated shift-invariant spaces. In case of the normal operators, we recover a diagonal decomposition. The results show, in particular, that any finitely generated shift-invariant space can be decomposed into an orthogonal sum of principal shift-invariant spaces, with additional invariance properties under a shift-preserving operator. Second, we provide a new characterization of the multi-tiling sets $Ω\subset\mathbb{R}^d$ of positive measure for which $L^2(Ω)$ admits a structured Riesz basis of exponentials that is formulated in the ambient space $\mathbb{T}^{k\times k}$. In addition, we show a simpler sufficient condition which generalizes the admissibility property, that is also necessary for 2-tiling sets.

Variations on two Cabrelli's works

TL;DR

The paper advances the theory of shift-invariant spaces by (i) deriving a triangular (and in the normal case, diagonal) decomposition for shift-preserving operators on finitely generated SIS via a fibered range-operator framework, and (ii) providing a practical ambient-space criterion in for when a multi-tiling set yields a structured Riesz basis of exponentials with periodic frequencies in . By reducing global questions to pointwise finite-dimensional problems on fibers, the authors unify operator canonical forms with multi-tiling spectral analysis and Paley-Wiener space structure, enabling explicit constructions and necessary/sufficient conditions grounded in determinant tests and a simple geometric separation property. The results generalize and sharpen previous decompositions (e.g., ACCP) and offer a tangible criterion for structured exponential bases in sampling and tiling contexts, with extensions to LCA groups. Overall, the work bridges shift-invariant theory, tiling, and Paley-Wiener analysis to yield actionable decompositions and basis characterizations for SIS and their spectral sets.

Abstract

In this paper we present two different problems within the framework of shift-invariant theory. First, we develop a triangular form for shift-preserving operators acting on finitely generated shift-invariant spaces. In case of the normal operators, we recover a diagonal decomposition. The results show, in particular, that any finitely generated shift-invariant space can be decomposed into an orthogonal sum of principal shift-invariant spaces, with additional invariance properties under a shift-preserving operator. Second, we provide a new characterization of the multi-tiling sets of positive measure for which admits a structured Riesz basis of exponentials that is formulated in the ambient space . In addition, we show a simpler sufficient condition which generalizes the admissibility property, that is also necessary for 2-tiling sets.
Paper Structure (9 sections, 20 theorems, 78 equations)

This paper contains 9 sections, 20 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Shift-invariant spaces
  3. Canonical forms for shift-preserving operators
  4. Shift-preserving operators
  5. Triangular form for shift-preserving operators
  6. Diagonal form for normal shift-preserving operators
  7. Bases of exponentials with periodic set of frequencies in Paley-Wiener spaces with multi-tiling spectrum
  8. Paley-Wiener spaces with multi-tiling spectrum
  9. Sufficient and necessary conditions for the existence of structured Riesz bases of exponentials

Key Result

Theorem 2.4

A closed subspace $V\subset L^{2}(\mathbb{R}^{d})$ is shift-invariant if and only if there exists a measurable range function such that Furthermore, if $V=S(\Phi)$ for some coun-table set $\Phi\subset L^2(\mathbb R^d)$, then for a.e. $\omega\in I$. Under the convention that two range functions are identified if they are equal a.e. $\omega\in I$, the correspondence between $V$ and $J$ is one-to-o

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: B
  • ...and 27 more