Cube tilings with linear constraints
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arXiv
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| Format: | Preprint |
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2024
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| author | Lee, Dae Gwan Pfander, Goetz E. Walnut, David |
| author_facet | Lee, Dae Gwan Pfander, Goetz E. Walnut, David |
| contents | We consider tilings $(\mathcal{Q},Φ)$ of $\mathbb{R}^d$ where $\mathcal{Q}$ is the $d$-dimensional unit cube and the set of translations $Φ$ is constrained to lie in a pre-determined lattice $A \mathbb{Z}^d$ in $\mathbb{R}^d$. We provide a full characterization of matrices $A$ for which such cube tilings exist when $Φ$ is a sublattice of $A\mathbb{Z}^d$ with any $d \in \mathbb{N}$ or a generic subset of $A\mathbb{Z}^d$ with $d\leq 7$. As a direct consequence of our results, we obtain a criterion for the existence of linearly constrained frequency sets, that is, $Φ\subseteq A\mathbb{Z}^d$, such that the respective set of complex exponential functions $\mathcal{E} (Φ)$ is an orthogonal Fourier basis for the space of square integrable functions supported on a parallelepiped $B\mathcal{Q}$, where $A, B \in \mathbb{R}^{d \times d}$ are nonsingular matrices given a priori. Similarly constructed Riesz bases are considered in a companion paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_07411 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cube tilings with linear constraints Lee, Dae Gwan Pfander, Goetz E. Walnut, David Classical Analysis and ODEs We consider tilings $(\mathcal{Q},Φ)$ of $\mathbb{R}^d$ where $\mathcal{Q}$ is the $d$-dimensional unit cube and the set of translations $Φ$ is constrained to lie in a pre-determined lattice $A \mathbb{Z}^d$ in $\mathbb{R}^d$. We provide a full characterization of matrices $A$ for which such cube tilings exist when $Φ$ is a sublattice of $A\mathbb{Z}^d$ with any $d \in \mathbb{N}$ or a generic subset of $A\mathbb{Z}^d$ with $d\leq 7$. As a direct consequence of our results, we obtain a criterion for the existence of linearly constrained frequency sets, that is, $Φ\subseteq A\mathbb{Z}^d$, such that the respective set of complex exponential functions $\mathcal{E} (Φ)$ is an orthogonal Fourier basis for the space of square integrable functions supported on a parallelepiped $B\mathcal{Q}$, where $A, B \in \mathbb{R}^{d \times d}$ are nonsingular matrices given a priori. Similarly constructed Riesz bases are considered in a companion paper. |
| title | Cube tilings with linear constraints |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2403.07411 |