Riesz bases of exponentials for multi-tiling measures
| Autor: | Lai, Chun-Kit, Sheynis, Alexander |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $G$ be a closed subgroup of ${\mathbb R}^d$ and let $\nu$ be a Borel probability measure admitting a Riesz basis of exponentials with frequency sets in the dual group $G^{\perp}$. We form a multi-tiling measure $\mu = \mu_1+...+\mu_N$ where $\mu_i$ is translationally equivalent to $\nu$ and different $\mu_i$ and $\mu_j$ have essentially disjoint support. We obtain some necessary and sufficient conditions for $\mu$ to admit a Riesz basis of exponentials . As an application, the square boundary, after a rotation, is a union of two fundamental domains of $G = {\mathbb Z}\times {\mathbb R}$ and can be regarded as a multi-tiling measure. We show that, unfortunately, the square boundary does not admit a Riesz basis of exponentials of the form as a union of translate of discrete subgroups ${\mathbb Z}\times \{0\}$. This rules out a natural candidate of potential Riesz basis for the square boundary. Comment: To appear in Sampling Theory, Signal Processing, and Data Analysis |
| Databáze: | arXiv |
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