Spectral structure and spectral eigenvalue problems of a class of self-similar spectral measures

He, Xing‐Gang; Tang, Min-Wei; Wu, Zhi-Yi

doi:10.1016/j.jfa.2019.05.019

Cited by 35 publications

(10 citation statements)

References 31 publications

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“…Many spectral measures have been found, see, for example [1,3,7,8,9,10,11,12,15,22,25,27,33,34,36,35,38] and references therein. For a singular spectral measure, generally speaking, its spectrum containing 0 is not unique, see [2,8,24,21,21,27,31,32] and references therein. However, it is a big challenge to determine all the spectra for a given singular spectral measure.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the quasi-Beurling dimensions of the spectra for planar Moran-type Sierpinski spectral measures

2023

Applied and Computational Harmonic Analysis

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the quasi-Beurling dimensions of the spectra for planar Moran-type Sierpinski spectral measures

2023

Applied and Computational Harmonic Analysis

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“…Dutkay and Haussermann [18] proved that if p is a prime number greater than 3, then p n Λ 1 is also a spectrum of µ 4,{0,2} for any n ≥ 1. He et al [25] gave an answer to the spectral eigenmatrix problems (1) about the measure µ q,{0,ar,br} for the spectrum Λ := ∞ k=1 {0, ±1}q k−1 . Fu et al [22] gave a complete characterization on the spectral eigenmatrix problems (2) of the Bernoulli convolution µ 2k .…”

Section: Introductionmentioning

confidence: 99%

“…He et al. [25] gave an answer to the spectral eigenmatrix problems (1) about the measure for the spectrum . Fu et al.…”

Section: Introductionmentioning

confidence: 99%

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

Liu,

Tang,

2023

Proceedings of the Edinburgh Mathematical Society

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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.

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“…At the same time, various new phenomena which are different from classic cases are discovered; see, e.g. [4,6,8,13,21,25,29,31,36,37]. For example, (i) there exists a maximal orthogonal set of exponentials of some self-similar measures is not necessarily an orthonormal basis [12,25]; (ii) there exist uncountable spectra for some singular spectral measures [21].…”

Section: Introductionmentioning

confidence: 99%

On spectral structure and spectral eigenvalue problems for a class of self similar spectral measure with product form

2022

Nonlinearity

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Let μ be a Borel probability measure with compact support on R d . We say that μ is a spectral measure if there exists Λ ⊆ R d , called a spectrum of μ, such that E ( Λ ) ≔ { e − 2 π i ⟨ λ , x ⟩ } λ ∈ Λ forms an orthonormal basis for L 2(μ). In this paper, we study the structure of spectra for a class of self-similar spectral measure μ R,B with product form on R . We first give a partially characterize for E Λ to be a maximal orthogonal family in L 2(μ R,B ) by using the notion of maximal tree mapping. Based on this, we give a sufficient condition for a maximal orthogonal family E Λ (which corresponds to a maximal tree mapping) to be an orthonormal basis of L 2(μ R,B ). Moreover, we completely settle two types of spectral eigenvalue problems for μ R,B . Precisely, on the first case, for the model spectrum (simplest spectrum) of μ R,B , we characterize all possible real numbers t such that tΛ is also a spectrum of μ R,B . On the other case, we characterize all possible real numbers t such that there exists a countable set Λ such that Λ and tΛ are both spectra of μ R,B .

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Spectral structure and spectral eigenvalue problems of a class of self-similar spectral measures

Cited by 35 publications

References 31 publications

On the quasi-Beurling dimensions of the spectra for planar Moran-type Sierpinski spectral measures

On the quasi-Beurling dimensions of the spectra for planar Moran-type Sierpinski spectral measures

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

On spectral structure and spectral eigenvalue problems for a class of self similar spectral measure with product form

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