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 $...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
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/ |
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| Summary: | 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]. |
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| ISSN: | 1778-3569 |