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Dai, Xin-Rong; Fu, Xiaoye; Yan, Zhi-Hui

doi:10.1016/j.acha.2019.12.001

Cited by 60 publications

(38 citation statements)

References 38 publications

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“…Let π x and π y be the canonical projections of R 2 onto the x and y-axes, respectively. The following proposition, which is proved in [12], establishes a relationship of orthogoanl sets between µ A,D and the self-similar measures µ 3q i ,{0,1,2} , i = 1, 2. Here, for any integer q ≥ 1, µ 3q,{0,1,2} is the unique Borel probability measure µ := µ 3q,{0,1,2} , which satisfies the invariance equation…”

Section: Beurling Dimension Of Spectra Of µ Admentioning

confidence: 83%

“…In [17], Deng and Lau made a start in studying the spectrality of µ A,D when A is a real expanding matrix with n = m ∈ R and they proved the only spectral Sierpiński measure are of n ∈ 3Z. Recently, Dai, Fu and Yan [12] completely characterized the spectrality of µ A,D when A is a real expanding matrix. They proved that µ A,D is a spectral measure if and only if n, m ∈ 3Z.…”

mentioning

confidence: 99%

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

2023

Ann. Funct. Anal.

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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.

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Section: Beurling Dimension Of Spectra Of µ Admentioning

confidence: 83%

mentioning

confidence: 99%

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

2023

Ann. Funct. Anal.

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“…Theorem 1. 4 Let D k , R k be given as in Theorem 1.2. Furthermore, we replace m k by m k r for k ≥ 2 where r is the rank of vectors {v, Rv, .…”

Section: Theorem 11 Suppose That Qmentioning

confidence: 99%

Spectrality of a class of Moran measures on with consecutive digit sets

Chen¹,

2022

Can. Math. Bull.

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Let $\{R_{k}\}_{k=1}^{\infty }$ be a sequence of expanding integer matrices in $M_{n}(\mathbb {Z})$ , and let $\{D_{k}\}_{k=1}^{\infty }$ be a sequence of finite digit sets with integer vectors in ${\mathbb Z}^{n}$ . In this paper, we prove that under certain conditions in terms of $(R_{k},D_{k})$ for $k\ge 1$ , the Moran measure $$ \begin{align*} \mu_{\{R_{k}\},\{D_{k}\}}:=\delta_{R_{1}^{-1}D_{1}}\ast\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}\ast\cdots \end{align*} $$ is a spectral measure. For the converse, we get a necessity condition for the admissible pair $(R,D)$ .

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“…In 1998, Jorgensen and Pedersen [24] gave the first singular spectral measure: the standard middle-fourth Cantor measure. Following these discoveries, many more examples of fractal spectral measures have been constructed, such as self-similar measures [4, 26], self-affine measures [11, 17, 31] and Moran measures [2, 3, 19]. It is surprising that there are many distinctive phenomena that singular spectral measures do have but the absolutely continuous ones do not.…”

Section: Introductionmentioning

confidence: 99%

“…If , the corresponding measure is a Sierpinski-type measure. Dai et al [17] and Deng et al [9] proved that is a spectral measure if and only if , . More recently, Wang [31] gave the analogous result of a class of spectral measure , where the digit set D satisfies certain conditions.…”

Section: Introductionmentioning

confidence: 99%

Spectrality of Moran Sierpinski-type measures on

Zhang

2021

Can. Math. Bull.

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Let $M=$ diag $(\rho _1,\rho _2)\in M_{2}({\mathbb R})$ be an expanding matrix and Let $\{D_n\}_{n=1}^{\infty }$ be a sequence of digit sets with $D_n=\left \{(0, 0)^T,\,\,\,(a_n, 0 )^T, \,\,\, (0, b_n )^T \right \}$ , where $a_n, b_n\in \{-1,1\}$ . The associated Borel probability measure $$ \begin{align*} \mu_{M,\{D_n\}}:=\delta_{M^{-1}D_1}\ast \delta_{M^{-2}D_2}\ast \delta_{M^{-3}D_3}\ast \cdots \end{align*} $$ is called a Moran Sierpinski-type measure. In this paper, we show that $\mu _{M, \{D_n\}}$ is a spectral measure if and only if $3\mid \rho _i$ for each $i=1, 2$ . The special case is the Sierpinski-type measure with $a_n=b_n=1$ for all $n\in {\mathbb N}$ , which is proved by Dai et al. [Appl. Comput. Harmon. Anal. (2020), https://doi.org/10.1016/j.acha.2019.12.001].

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Spectrality of self-affine Sierpinski-type measures on R2

Cited by 60 publications

References 38 publications

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

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

Spectrality of a class of Moran measures on with consecutive digit sets

Spectrality of Moran Sierpinski-type measures on

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