The generalized Fuglede's conjecture holds for a class of Cantor-Moran measures
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912227483713536 |
|---|---|
| author | An, Lixiang Li, Qian Zhang, Minmin |
| author_facet | An, Lixiang Li, Qian Zhang, Minmin |
| contents | Suppose ${\bf b}=\{b_n\}_{n=1}^{\infty}$ is a sequence of integers bigger than 1 and ${\bf D}=\{{\mathcal D}_{n}\}_{n=1}^{\infty}$ is a sequence of consecutive digit sets. Let $μ_{{\bf b},{\bf D}}$ be the Cantor-Moran measure defined by \begin{eqnarray*}
μ_{{\bf b},{\bf D}}&=& δ_{\frac{1}{b_1}{\mathcal D}_{1}}\astδ_{\frac{1}{b_1b_2}{\mathcal D}_{2}}\ast δ_{\frac{1}{b_1b_2b_3}{\mathcal D}_{3}}\ast\cdots.
\end{eqnarray*}
We prove that $L^2(μ_{{\bf b},{\bf D}})$ possesses an exponential orthonormal basis if and only if $μ_{{\bf b},{\bf D}}\astν={\mathcal L}_{[0,N_1/b_1]}$ for some Borel probability measure $ν$.
This theorem shows that the generalized Fuglede's conjecture is true for such Cantor-Moran measure. An immediate consequence of this result is the equivalence between the existence of an exponential orthonormal basis and the integral tiling of ${\bf D}_n={\mathcal D}_{n}+b_n{\mathcal D}_{n-1}+b_2\cdots b_n{\mathcal D}_{1}$ for $n\geq1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_12738 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The generalized Fuglede's conjecture holds for a class of Cantor-Moran measures An, Lixiang Li, Qian Zhang, Minmin Functional Analysis 42C05, 46C05, 28A80 Suppose ${\bf b}=\{b_n\}_{n=1}^{\infty}$ is a sequence of integers bigger than 1 and ${\bf D}=\{{\mathcal D}_{n}\}_{n=1}^{\infty}$ is a sequence of consecutive digit sets. Let $μ_{{\bf b},{\bf D}}$ be the Cantor-Moran measure defined by \begin{eqnarray*} μ_{{\bf b},{\bf D}}&=& δ_{\frac{1}{b_1}{\mathcal D}_{1}}\astδ_{\frac{1}{b_1b_2}{\mathcal D}_{2}}\ast δ_{\frac{1}{b_1b_2b_3}{\mathcal D}_{3}}\ast\cdots. \end{eqnarray*} We prove that $L^2(μ_{{\bf b},{\bf D}})$ possesses an exponential orthonormal basis if and only if $μ_{{\bf b},{\bf D}}\astν={\mathcal L}_{[0,N_1/b_1]}$ for some Borel probability measure $ν$. This theorem shows that the generalized Fuglede's conjecture is true for such Cantor-Moran measure. An immediate consequence of this result is the equivalence between the existence of an exponential orthonormal basis and the integral tiling of ${\bf D}_n={\mathcal D}_{n}+b_n{\mathcal D}_{n-1}+b_2\cdots b_n{\mathcal D}_{1}$ for $n\geq1$. |
| title | The generalized Fuglede's conjecture holds for a class of Cantor-Moran measures |
| topic | Functional Analysis 42C05, 46C05, 28A80 |
| url | https://arxiv.org/abs/2405.12738 |