Variations on two Cabrelli's works

Elona Agora; Jorge Antezana; Diana Carbajal

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arXiv · 2601.05422

Variations on two Cabrelli's works

Elona Agora·Jorge Antezana·Diana Carbajal

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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 $\Omega\subset\mathbb{R}^d$ of positive measure for which $L^2(\Omega)$ 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.

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Elona Agora, Jorge Antezana, Diana Carbajal. 2026-01-08. Variations on two Cabrelli's works. https://arxiv.org/abs/2601.05422

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