Spectral properties of a class of Sierpinski-type Moran measures on $ \mathbb{R}^n $
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912567975215104 |
|---|---|
| author | Chen, Jia Long Yan, Xiao-Yu |
| author_facet | Chen, Jia Long Yan, Xiao-Yu |
| contents | Let the infinite convolutions \begin{equation*}
μ_{\{R_{k}\},\{D_{k}\}}=δ_{R_{1}^{-1}D_{1}}*δ_{R_{1}^{-1}R_{2}^{-1}D_{2}}*δ_{R_{1}^{-1}R_{2}^{-1}R_{3}^{-1}D_{3}}*\dotsi \end{equation*}
be generated by the sequence of pairs \(\{ (R_k,D_k) \}_{k=1}^{\infty} \), where $ R_k\in M_n(\mathbb{Z})$ is an expanding integer matric, $D_k$ is a finite integer digit sets that satisfies the following two conditions: (i). \( \# D_k = m \) and \( m>2 \) is a prime; (ii). \( \{x: \sum_{d\in D_{k}}e^{2πi\langle d,x \rangle}=0\} =\cup_{i=1}^{ϕ(k)}\cup_{j=1}^{m-1}(\frac{j}{m}ν_{k,i}+\mathbb{Z}^{n}) \) for some \( ν_{k,i} \in \{ (l_1, \cdots, l_n)^t : l_i \in [1, m-1] \cap \mathbb{Z}, 1\leq i\leq n \} \). In this paper, we study the spectrality of $μ_{\{R_{k}\},\{D_{k}\}}$, and some necessary and sufficient conditions for \( L^{2}(μ_{\{R_{k}\},\{D_{k}\}}) \) to have an orthogonal exponential function basis are established. Finally, we discuss the explanations and applications of our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09360 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral properties of a class of Sierpinski-type Moran measures on $ \mathbb{R}^n $ Chen, Jia Long Yan, Xiao-Yu Classical Analysis and ODEs Dynamical Systems 28A80, 42C05, 46C05 Let the infinite convolutions \begin{equation*} μ_{\{R_{k}\},\{D_{k}\}}=δ_{R_{1}^{-1}D_{1}}*δ_{R_{1}^{-1}R_{2}^{-1}D_{2}}*δ_{R_{1}^{-1}R_{2}^{-1}R_{3}^{-1}D_{3}}*\dotsi \end{equation*} be generated by the sequence of pairs \(\{ (R_k,D_k) \}_{k=1}^{\infty} \), where $ R_k\in M_n(\mathbb{Z})$ is an expanding integer matric, $D_k$ is a finite integer digit sets that satisfies the following two conditions: (i). \( \# D_k = m \) and \( m>2 \) is a prime; (ii). \( \{x: \sum_{d\in D_{k}}e^{2πi\langle d,x \rangle}=0\} =\cup_{i=1}^{ϕ(k)}\cup_{j=1}^{m-1}(\frac{j}{m}ν_{k,i}+\mathbb{Z}^{n}) \) for some \( ν_{k,i} \in \{ (l_1, \cdots, l_n)^t : l_i \in [1, m-1] \cap \mathbb{Z}, 1\leq i\leq n \} \). In this paper, we study the spectrality of $μ_{\{R_{k}\},\{D_{k}\}}$, and some necessary and sufficient conditions for \( L^{2}(μ_{\{R_{k}\},\{D_{k}\}}) \) to have an orthogonal exponential function basis are established. Finally, we discuss the explanations and applications of our results. |
| title | Spectral properties of a class of Sierpinski-type Moran measures on $ \mathbb{R}^n $ |
| topic | Classical Analysis and ODEs Dynamical Systems 28A80, 42C05, 46C05 |
| url | https://arxiv.org/abs/2505.09360 |