The spectrality of symmetric additive measures

Wen-Hui Ai; Zheng-Yi Lu; Ting Zhou

doi:10.5802/crmath.435

Harmonic analysis

The spectrality of symmetric additive measures

Wen-Hui Ai ¹; Zheng-Yi Lu ¹ ; Ting Zhou ²

¹Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China
²School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P. R. China

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 783-793

Abstract

Let $ρ$ be a symmetric measure of Lebesgue type, i.e.,

ρ = \frac{1}{2} (μ \times δ_{0} + δ_{0} \times μ),

where the component measure $μ$ is the Lebesgue measure supported on $[t, t + 1]$ for $t \in ℚ ∖ {- \frac{1}{2}}$ and $δ_{0}$ is the Dirac measure at $0$ . We prove that $ρ$ is a spectral measure if and only if $t \in \frac{1}{2} ℤ$ . In this case, $L^{2} (ρ)$ has a unique orthonormal basis of the form

\{e^{2 π i (λ x - λ y)} : λ \in Λ_{0}\},

where $Λ_{0}$ 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].

Received: 2022-06-13
Accepted: 2022-10-10
Published online: 2023-05-11

DOI: 10.5802/crmath.435

Classification: 28A80, 42C05

Author's affiliations:

Wen-Hui Ai ¹ ; Zheng-Yi Lu ¹ ; Ting Zhou ²

¹ Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China
² School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P. R. China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Wen-Hui Ai and Zheng-Yi Lu and Ting Zhou},
     title = {The spectrality of symmetric additive measures},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {783--793},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     doi = {10.5802/crmath.435},
     language = {en},
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Wen-Hui Ai; Zheng-Yi Lu; Ting Zhou. The spectrality of symmetric additive measures. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 783-793. doi: 10.5802/crmath.435

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