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Spectrality of a Class of Moran Measures

Part of: Nontrigonometric harmonic analysis Normed linear spaces and Banach spaces; Banach lattices Classical measure theory

Published online by Cambridge University Press:  17 January 2020

Ming-Liang Chen ,
Jing-Cheng Liu ,
Juan Su  and
Xiang-Yang Wang
Ming-Liang Chen
Affiliation:
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P.R. China Email: mathcml@163.commcswxy@mail.sysu.edu.cn
Jing-Cheng Liu
Affiliation:
Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P.R. China Email: jcliu@hunnu.edu.cn
Juan Su
Affiliation:
Changsha School of Mathematics and Statistics, University of Science & Technology, Changsha, Hunan 410114, P.R. China Email: suj1234@126.com
Xiang-Yang Wang
Affiliation:
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P.R. China Email: mathcml@163.commcswxy@mail.sysu.edu.cn
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Abstract

Let $\{M_{n}\}_{n=1}^{\infty }$ be a sequence of expanding matrices with $M_{n}=\operatorname{diag}(p_{n},q_{n})$, and let $\{{\mathcal{D}}_{n}\}_{n=1}^{\infty }$ be a sequence of digit sets with ${\mathcal{D}}_{n}=\{(0,0)^{t},(a_{n},0)^{t},(0,b_{n})^{t},\pm (a_{n},b_{n})^{t}\}$, where $p_{n}$, $q_{n}$, $a_{n}$ and $b_{n}$ are positive integers for all $n\geqslant 1$. If $\sup _{n\geqslant 1}\{\frac{a_{n}}{p_{n}},\frac{b_{n}}{q_{n}}\}<\infty$, then the infinite convolution $\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$ is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set $\unicode[STIX]{x1D6EC}$ such that $\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$ is an orthonormal basis for $L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$.

Keywords

Cantor–Dust–Moran measurespectral measurespectruminfinite convolution

MSC classification

Primary: 28A80: Fractals
Secondary: 42C05: Orthogonal functions and polynomials, general theory 46C05: Hilbert and pre-Hilbert spaces
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 366 - 381
Copyright
© Canadian Mathematical Society 2019

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Footnotes

This research is supported in part by the NNSF of China (Nos. 11401053, 11771457, 11571104 and 11971500) and the SRF of Hunan Provincial Education Department (Nos. 17B158 and 14C0046). Author X.-Y. W. is the corresponding author.

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