Spectrality and non-spectrality of the Riesz product measures with three elements in digit sets

An, Li-Xiang; He, Liu; He, Xing‐Gang

doi:10.1016/j.jfa.2018.10.017

Cited by 37 publications

(14 citation statements)

References 29 publications

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“…Many affirmative results have been obtained in [1][2][3][4][5]11,15,19,27,32]. When all Hadamard triples are the same one, the infinite convolution reduces to a self-affine measure (which is called a self-similar measure for d = 1).…”

Section: Question Given a Sequence Of Hadamard Triplesmentioning

confidence: 97%

Spectrality of random convolutions generated by finitely many Hadamard triples

Li¹,

Miao²,

Wang³

2022

Preprint

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Section: Question Given a Sequence Of Hadamard Triplesmentioning

confidence: 97%

Spectrality of random convolutions generated by finitely many Hadamard triples

Li¹,

Miao²,

Wang³

2022

Preprint

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“…Thus, we have that is a spectrum of and is an orthogonal set of . The following lemma is motivated by Theorem 2.3 in [2] and easy to prove.…”

Section: Preliminaries and The Sufficiency Of Theorem 11mentioning

confidence: 97%

“…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%

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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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“…Many affirmative partial results have been obtained for this question, see [1][2][3][4][5]18,20,25,35]. If all Hadamard triples (R k , B k , L k ) = (R, B, L), then the infinite convolution reduces to a self-affine measure.…”

Section: Given a Sequence Of Hadamard Triplesmentioning

confidence: 99%

Weak Convergence and Spectrality of Infinite Convolutions

Li¹,

Miao²,

Wang³

2022

Preprint

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Let {A k } ∞ k=1 be a sequence of finite subsets of R d satisfying that #A k ≥ 2 for all integers k ≥ 1. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolutiona∈A δ a . Then we study the spectrality of a class of infinite convolutions generated by Hadamard triples in R and construct a class of singular spectral measures without compact support. Finally we show that such measures are abundant, and the dimension of their supports has the intermediate-value property.

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Spectrality and non-spectrality of the Riesz product measures with three elements in digit sets

Cited by 37 publications

References 29 publications

Spectrality of random convolutions generated by finitely many Hadamard triples

Spectrality of random convolutions generated by finitely many Hadamard triples

Spectrality of Moran Sierpinski-type measures on

Weak Convergence and Spectrality of Infinite Convolutions

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