1. The spectrality of symmetric additive measures
- Author
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Ai, Wen-Hui, Lu, Zheng-Yi, and Zhou, Ting
- Subjects
Mathematics ,QA1-939 - Abstract
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].
- Published
- 2023
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