On the spectrality of a class of Moran measures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914783456919552 |
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| author | Zheng, Yali Xiao, Yingqing |
| author_facet | Zheng, Yali Xiao, Yingqing |
| contents | In this paper, we study the spectrality of a class of Moran measures $μ_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n,\mathcal{D}_n)\}_{n=1}^{\infty}$, where $\mathcal{P}=\{p_n\}_{n=1}^{\infty}$ is a sequence of positive integers with $p_n>1$ and $\mathcal{D}=\{\mathcal{D}_{n}\}_{n=1}^{\infty}$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality $\#\mathcal{D}_{n}\in \{2,3,N_{n}\}$. We find a countable set $Λ\subset\mathbb{R}$ such that the set $\{e^{-2πi λx}|λ\inΛ\}$ is a orthonormal basis of $L^{2}(μ_{\mathcal{P},\mathcal{D}})$ under some conditions. As an application, we show that when $μ_{\mathcal{P},\mathcal{D}}$ is absolutely continuous, $μ_{\mathcal{P},\mathcal{D}}$ not only is a spectral measure, but also its support set tiles $\mathbb{R}$ with $\mathbb{Z}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_18690 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the spectrality of a class of Moran measures Zheng, Yali Xiao, Yingqing Classical Analysis and ODEs 28A80, 28A25, 42C05, 46C05 In this paper, we study the spectrality of a class of Moran measures $μ_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n,\mathcal{D}_n)\}_{n=1}^{\infty}$, where $\mathcal{P}=\{p_n\}_{n=1}^{\infty}$ is a sequence of positive integers with $p_n>1$ and $\mathcal{D}=\{\mathcal{D}_{n}\}_{n=1}^{\infty}$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality $\#\mathcal{D}_{n}\in \{2,3,N_{n}\}$. We find a countable set $Λ\subset\mathbb{R}$ such that the set $\{e^{-2πi λx}|λ\inΛ\}$ is a orthonormal basis of $L^{2}(μ_{\mathcal{P},\mathcal{D}})$ under some conditions. As an application, we show that when $μ_{\mathcal{P},\mathcal{D}}$ is absolutely continuous, $μ_{\mathcal{P},\mathcal{D}}$ not only is a spectral measure, but also its support set tiles $\mathbb{R}$ with $\mathbb{Z}$. |
| title | On the spectrality of a class of Moran measures |
| topic | Classical Analysis and ODEs 28A80, 28A25, 42C05, 46C05 |
| url | https://arxiv.org/abs/2404.18690 |