On Spectral Eigenvalue Problem of a Class of Self-similar Spectral Measures with Consecutive Digits

Wu, Zhi-Yi

doi:10.1007/s00041-020-09795-x

Cited by 7 publications

(3 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[22] gave a complete characterization on the spectral eigenmatrix problems (2) of the Bernoulli convolution . Wang and Wu [45] studied the spectral eigenmatrix problems (2) for a class of self-similar spectral measures with consecutive digits. The known results on spectral eigenvalue problems mainly focus on self-similar spectral measures in one-dimensional case; however, there are not much discussions about the spectral eigenmatrix problems in high-dimensional case since the methods in one dimension are difficult to apply to higher dimensions even for the simple cases.…”

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

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

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

View full text Add to dashboard Cite

show abstract

“…In high dimension, very few results are known except that the self-similar Sierpinski type spectral measure has been investigated and settled completely by An et al [1]. More general spectral eigenvalue problem for a class of self-similar spectral measures with consecutive digits has been studied by Wang and the second author [39].…”

Section: Introductionmentioning

confidence: 99%

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

2022

Nonlinearity

View full text Add to dashboard Cite

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 .

show abstract

“…This is rather surprising because the scaled set becomes more sparse but keeps the completeness. The exotic phenomenon naturally yields the spectral eigenvalue problem [27,29,11,31,38,45] and further spectral structure problem [21,13,11,29,15,3].…”

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

On Spectral Eigenvalue Problem of a Class of Self-similar Spectral Measures with Consecutive Digits

Cited by 7 publications

References 30 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

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

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

Contact Info

Product

Resources

About