Abstract
Let \(\mu \) be a Borel probability measure on \({\mathbb {R}}^n\). We call \(\mu \) a spectral measure if there exists a countable set \(\Lambda \subset {\mathbb {R}}^n\) such that \(E_\Lambda :=\{e^{2\pi i<\lambda ,x>}:\lambda \in \Lambda \}\) forms an orthogonal basis for the Hilbert space \(L^2(\mu )\). Let the measure \(\mu _{\{M,{\mathcal {D}}_n\}}\) be defined by the following expression \(\mu _{\{M,{\mathcal {D}}_n\}}=\delta _{M^{-1}{\mathcal {D}}_1}*\delta _{M^{-2}{\mathcal {D}}_2}*\cdots \), where \(M=\text {diag}(\rho ^{-1},\rho ^{-1})\) with \(|\rho |<1\), and \({\mathcal {D}}_n=\left\{ (0,0)^t,(a_n,b_n)^t,(c_n,d_n)^t)\right\} \) with \(|a_n d_n-b_n c_n|=1\) for all \(n\ge 1\). This paper focuses on the spectrality of a class of Moran measures \(\mu _{\{M,{\mathcal {D}}_n\}}\) in \({\mathbb {R}}^2\). We give some sufficient conditions for \(\mu _{\{M,{\mathcal {D}}_n\}}\) to be a spectral measure. Also some necessary conditions are obtained. Moreover, the spectrality of the above Moran measures are also used to determine the spectrality of certain self-affine measures.
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Communicated by H. Bruin.
Dedicated to the memory of Professor Tian-You Hu.
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The research is supported in part by the NNSF of China (Nos. 11831007 and 12071125) and Scientific Research Fund of Hunan Provincial Education Department (No. 20B386). The author Zheng-Yi Lu is also supported by Hunan Provincial Innovation Foundation for Postgraduate (CX20200518).
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Lu, ZY., Dong, XH. Spectrality of a class of Moran measures. Monatsh Math 196, 207–230 (2021). https://doi.org/10.1007/s00605-021-01558-0
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DOI: https://doi.org/10.1007/s00605-021-01558-0