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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2511.21341 |
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| _version_ | 1866908677060952064 |
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| author | Yan, Xiao-Yu Ai, Wen-Hui |
| author_facet | Yan, Xiao-Yu Ai, Wen-Hui |
| contents | A Borel probability measure \( μ\) with compact support on \( \mathbb{R}^n \) is called spectral measure if there exists a discrete set \( Λ\subset \mathbb{R}^n \) such that \( E_Λ:= \{e^{2πi \langle λ, x \rangle}: λ\in Λ\} \) forms an orthonormal basis of \( L^2(μ) \). In this paper, we study the spectrality and non-spectrality of a class of Moran measures with three-element digits on \( \mathbb{R} \). Let $p_n\in 3\mathbb Z\setminus\{0\}$ and $\mathcal{D}_n=\{0,a_n,b_n\}$ with $\{a_n,b_n\}=\{-1,1\}\pmod 3$. It is know that the infinite convolution of uniformly discrete probability measures $$μ_{\{p_n\},\{\mathcal D_n\}}:=δ_{p_1^{-1}\{0,a_1,b_1\}}\astδ_{(p_1p_2)^{-1}\{0,a_2,b_2\}}\ast\cdots $$ is a Moran measure with compact support if and only if \begin{align*} \sum_{n=1}^{\infty}|p_{1}p_{2}\cdots p_n|^{-1}d_n<\infty,\quad \mbox{where}\;d_n=\max\{0,|a_n|, |b_n|\}. \end{align*} Without the condition $\sup_{n\geq 1}\{\frac{|a_n|+|b_n|}{|p_n|}\}<\infty$, we give two sufficient conditions under which that $μ_{\{p_n\},\{\mathcal D_n\}}$ is a spectral measure. If $p_n=p>2$ and $\mathcal{D}_n=\{0,a_n,b_n\}$ with $\gcd(a_n,b_n)=1$, we also find an useful condition to guarantee that $μ_{p,\{\mathcal D_n\}}$ is not a spectral measure. Our results extend some known theorems in An et al. [JFA, 2019] and Lu et al. [JFAA, 2022]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21341 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectrality and non-spectrality of a class of Moran measures with three-element digits Yan, Xiao-Yu Ai, Wen-Hui Functional Analysis A Borel probability measure \( μ\) with compact support on \( \mathbb{R}^n \) is called spectral measure if there exists a discrete set \( Λ\subset \mathbb{R}^n \) such that \( E_Λ:= \{e^{2πi \langle λ, x \rangle}: λ\in Λ\} \) forms an orthonormal basis of \( L^2(μ) \). In this paper, we study the spectrality and non-spectrality of a class of Moran measures with three-element digits on \( \mathbb{R} \). Let $p_n\in 3\mathbb Z\setminus\{0\}$ and $\mathcal{D}_n=\{0,a_n,b_n\}$ with $\{a_n,b_n\}=\{-1,1\}\pmod 3$. It is know that the infinite convolution of uniformly discrete probability measures $$μ_{\{p_n\},\{\mathcal D_n\}}:=δ_{p_1^{-1}\{0,a_1,b_1\}}\astδ_{(p_1p_2)^{-1}\{0,a_2,b_2\}}\ast\cdots $$ is a Moran measure with compact support if and only if \begin{align*} \sum_{n=1}^{\infty}|p_{1}p_{2}\cdots p_n|^{-1}d_n<\infty,\quad \mbox{where}\;d_n=\max\{0,|a_n|, |b_n|\}. \end{align*} Without the condition $\sup_{n\geq 1}\{\frac{|a_n|+|b_n|}{|p_n|}\}<\infty$, we give two sufficient conditions under which that $μ_{\{p_n\},\{\mathcal D_n\}}$ is a spectral measure. If $p_n=p>2$ and $\mathcal{D}_n=\{0,a_n,b_n\}$ with $\gcd(a_n,b_n)=1$, we also find an useful condition to guarantee that $μ_{p,\{\mathcal D_n\}}$ is not a spectral measure. Our results extend some known theorems in An et al. [JFA, 2019] and Lu et al. [JFAA, 2022]. |
| title | Spectrality and non-spectrality of a class of Moran measures with three-element digits |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2511.21341 |