The spectrality of symmetric additive measures
Let $\rho $ be a symmetric measure of Lebesgue type, i.e., \begin{equation*} \rho =\frac{1}{2}(\mu \times \delta _0+\delta _0\times \mu ), \end{equation*} where the component measure $\mu $ is the Lebesgue measure supported on $[t, t+1]$ for $t\in \mathbb{Q}\setminus \lbrace -\frac{1}{2}\rbrace $...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.435/ |
| _version_ | 1797651508074905600 |
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| author | Ai, Wen-Hui Lu, Zheng-Yi Zhou, Ting |
| author_facet | Ai, Wen-Hui Lu, Zheng-Yi Zhou, Ting |
| author_sort | Ai, Wen-Hui |
| collection | DOAJ |
| description | Let $\rho $ be a symmetric measure of Lebesgue type, i.e.,
\begin{equation*}
\rho =\frac{1}{2}(\mu \times \delta _0+\delta _0\times \mu ),
\end{equation*}
where the component measure $\mu $ is the Lebesgue measure supported on $[t, t+1]$ for $t\in \mathbb{Q}\setminus \lbrace -\frac{1}{2}\rbrace $ and $\delta _{0}$ is the Dirac measure at $0$. We prove that $\rho $ is a spectral measure if and only if $ t \in \frac{1}{2}\mathbb{Z}$. In this case, $L^2(\rho )$ has a unique orthonormal basis of the form
\[ \left\lbrace e^{2\pi i (\lambda x-\lambda y)}:\lambda \in \Lambda _0\right\rbrace , \]
where $\Lambda _0$ is the spectrum of the Lebesgue measure supported on $[-t-1,-t]\cup [t,t+1]$. Our result answers some questions raised by Lai, Liu and Prince [JFA, 2021]. |
| first_indexed | 2024-03-11T16:16:47Z |
| format | Article |
| id | doaj.art-f133b2ad629746fa9870222212e31833 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:47Z |
| publishDate | 2023-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-f133b2ad629746fa9870222212e318332023-10-24T14:20:34ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-05-01361G478379310.5802/crmath.43510.5802/crmath.435The spectrality of symmetric additive measuresAi, Wen-Hui0Lu, Zheng-Yi1Zhou, Ting2Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaKey Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaSchool of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P. R. ChinaLet $\rho $ be a symmetric measure of Lebesgue type, i.e., \begin{equation*} \rho =\frac{1}{2}(\mu \times \delta _0+\delta _0\times \mu ), \end{equation*} where the component measure $\mu $ is the Lebesgue measure supported on $[t, t+1]$ for $t\in \mathbb{Q}\setminus \lbrace -\frac{1}{2}\rbrace $ and $\delta _{0}$ is the Dirac measure at $0$. We prove that $\rho $ is a spectral measure if and only if $ t \in \frac{1}{2}\mathbb{Z}$. In this case, $L^2(\rho )$ has a unique orthonormal basis of the form \[ \left\lbrace e^{2\pi i (\lambda x-\lambda y)}:\lambda \in \Lambda _0\right\rbrace , \] where $\Lambda _0$ is the spectrum of the Lebesgue measure supported on $[-t-1,-t]\cup [t,t+1]$. Our result answers some questions raised by Lai, Liu and Prince [JFA, 2021].https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.435/ |
| spellingShingle | Ai, Wen-Hui Lu, Zheng-Yi Zhou, Ting The spectrality of symmetric additive measures Comptes Rendus. Mathématique |
| title | The spectrality of symmetric additive measures |
| title_full | The spectrality of symmetric additive measures |
| title_fullStr | The spectrality of symmetric additive measures |
| title_full_unstemmed | The spectrality of symmetric additive measures |
| title_short | The spectrality of symmetric additive measures |
| title_sort | spectrality of symmetric additive measures |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.435/ |
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