Spectrality of a class of self-affine measures on 
	    
	    R2
	    
	  
               *

Chen, Mingliang; Liu, Jingcheng; Wang, Xiangyang

doi:10.1088/1361-6544/ac2493

Cited by 14 publications

(30 citation statements)

References 42 publications

(44 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are many researches about its spectrality or non-spectrality [35,37,42,43]. Recently, Chen et al [5] gave the following complete characterization on the spectrality of µ M,D .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The spectral eigenmatrix problems of planar self-affine measures with four digits

Liu,

Tang,

2023

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

Given a Borel probability measure µ on $\mathbb{R}^n$ and a real matrix $R\in M_n(\mathbb{R})$. We call R a spectral eigenmatrix of the measure µ if there exists a countable set $\Lambda\subset \mathbb{R}^n$ such that the sets $E_\Lambda=\big\{{\rm e}^{2\pi i \langle\lambda,x\rangle}:\lambda\in \Lambda\big\}$ and $E_{R\Lambda}=\big\{{\rm e}^{2\pi i \langle R\lambda,x\rangle}:\lambda\in \Lambda\big\}$ are both orthonormal bases for the Hilbert space $L^2(\mu)$. In this paper, we study the structure of spectral eigenmatrix of the planar self-affine measure $\mu_{M,D}$ generated by an expanding integer matrix $M\in M_2(2\mathbb{Z})$ and the four-elements digit set $D = \{(0,0)^t,(1,0)^t,(0,1)^t,(-1,-1)^t\}$. Some sufficient and/or necessary conditions for R to be a spectral eigenmatrix of $\mu_{M,D}$ are given.

show abstract

“…There are many researches about its spectrality or non-spectrality [35,37,42,43]. Recently, Chen et al [5] gave the following complete characterization on the spectrality of µ M,D .…”

Section: Introductionmentioning

confidence: 99%

“…[5] Let be an integer digit set and be an expanding matrix. If , then is a spectral measure if and only if .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The spectral eigenmatrix problems of planar self-affine measures with four digits

Liu,

Tang,

2023

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

show abstract

“…In 1998, Jorgensen and Pedersen [ 2 ] discovered the first singular, non-atomic spectral measure—the middle-forth Cantor measure—and proved the middle-third Cantor measure is not a spectral measure. Following this discovery, there has been much research on the spectrality of self-similar (or self-affine) measures and Moran-type self-similar (or self-affine) measures (see for example [ 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 ] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Spectra of Self-Similar Measures

Cao

Deng

2022

Entropy

View full text Add to dashboard Cite

This paper is devoted to the characterization of spectrum candidates with a new tree structure to be the spectra of a spectral self-similar measure μN,D generated by the finite integer digit set D and the compression ratio N−1. The tree structure is introduced with the language of symbolic space and widens the field of spectrum candidates. The spectrum candidate considered by aba and Wang is a set with a special tree structure. After showing a new criterion for the spectrum candidate with a tree structure to be a spectrum of μN,D, three sufficient and necessary conditions for the spectrum candidate with a tree structure to be a spectrum of μN,D were obtained. This result extends the conclusion of aba and Wang. As an application, an example of spectrum candidate Λ(N,B) with the tree structure associated with a self-similar measure is given. By our results, we obtain that Λ(N,B) is a spectrum of the self-similar measure. However, neither the method of aba and Wang nor that of Strichartz is applicable to the set Λ(N,B).

show abstract

“…In 1998, Jorgensen and Pederson [32] showed that the standard middlefourth Cantor measure µ 4,{0,2} is a spectral measure, which marks the entrance of Fourier analysis into the realm of fractals. Since then, much work has been devoted to studying the spectrality of self-similar measures, selfaffine measures and Moran measures, see [2,4,6,8,10,16,14,19,17,42,22,35,34,33,41] and references therein. Around the same time, various new phenomena different from spectral theory for the Lebesgue measure have been found.…”

mentioning

confidence: 99%

Beurling dimension of a class of spectra of the Sierpinski-type spectral measures

2023

Ann. Funct. Anal.

View full text Add to dashboard Cite

In this paper, we study the structure of the spectra for the Sierpiński type spectral measure µA,D on R 2 . We give a sufficient and necessary condition for the family of exponential functions {e −2πi λ,x : λ ∈ Λ} to be a maximal orthogonal set in L 2 (µA,D). Based on this result, we obtain a class of regular spectra of µA,D. Moreover, we discuss the Beurling dimensions of the spectra and obtain the optimal upper bound of Beurling dimensions of all spectra, which is in stark contrast with the case of self-similar spectral measure. An intermediate property about the Beurling dimension of the spectra is obtained.

show abstract

Spectrality of a class of self-affine measures on R2 *

Abstract: In this work, we study the spectral property of a class of self-affine measures μ M,D on R 2 generated by an expanding real ma… Show more

Cited by 14 publications

References 42 publications

The spectral eigenmatrix problems of planar self-affine measures with four digits

The spectral eigenmatrix problems of planar self-affine measures with four digits

Spectra of Self-Similar Measures

Beurling dimension of a class of spectra of the Sierpinski-type spectral measures

Contact Info

Product

Resources

About