Variations on two Cabrelli's works
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917191416283136 |
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| author | Agora, Elona Antezana, Jorge Carbajal, Diana |
| author_facet | Agora, Elona Antezana, Jorge Carbajal, Diana |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05422 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Variations on two Cabrelli's works Agora, Elona Antezana, Jorge Carbajal, Diana Functional Analysis Classical Analysis and ODEs 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. |
| title | Variations on two Cabrelli's works |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2601.05422 |