On self-similar spectral measures

An, Li-Xiang; Wang, Cong

doi:10.1016/j.jfa.2020.108821

Cited by 25 publications

(11 citation statements)

References 31 publications

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“…They showed that the one-fourth Cantor measure is a spectral measure, but the one-third Cantor measure is not. Further research on the spectrality and nonspectrality of measures treats self-similar measures (see [2] for a recent example), self-affine measures (see [4,5]) and Moran measures (see [1]). [2] Planar Moran-Sierpinski measures 309…”

Section: Introductionmentioning

confidence: 99%

“…), where Λ (1) is the first coordinate of Λ and Λ (2) is the second coordinate. By the orthogonality of Λ, we have (Λ − Λ) \ {0} ⊂ Z( μ M,{D n } ).…”

Section: Introductionmentioning

confidence: 99%

“…We now claim that Λ (i) is infinite for i = 1, 2. It is enough to prove that Λ (1) is infinite, since the proof for Λ (2) is similar. Suppose to the contrary that Λ (1) is finite.…”

Section: Introductionmentioning

confidence: 99%

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A Nonspectral Problem for Planar Moran–sierpinski Measures

Wei²

2023

Bull. Aust. Math. Soc.

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Let $M=(\begin {smallmatrix}\rho ^{-1} & 0 \\0 & \rho ^{-1} \\\end {smallmatrix})$ be an expanding real matrix with $0<\rho <1$ , and let ${\mathcal D}_n=\{(\begin {smallmatrix} 0\\ 0 \end {smallmatrix}),(\begin {smallmatrix} \sigma _n\\ 0 \end {smallmatrix}),(\begin {smallmatrix} 0\\ \gamma _n \end {smallmatrix})\}$ be digit sets with $\sigma _n,\gamma _n\in \{-1,1\}$ for each $n\ge 1$ . Then the infinite convolution $$ \begin{align*}\mu_{M,\{{\mathcal D}_n\}}=\delta_{M^{-1}{\mathcal D}_1}\ast\delta_{M^{-2}{\mathcal D}_2}\ast\cdots\end{align*} $$ is called a Moran–Sierpinski measure. We give a necessary and sufficient condition for $L^2(\,\mu _{M,\{{\mathcal D}_n\}})$ to admit an infinite orthogonal set of exponential functions. Furthermore, we give the exact cardinality of orthogonal exponential functions in $L^2(\,\mu _{M,\{{\mathcal D}_n\}})$ when $L^2(\,\mu _{M,\{{\mathcal D}_n\}})$ does not admit any infinite orthogonal set of exponential functions based on whether $\rho $ is a trinomial number or not.

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Section: Introductionmentioning

confidence: 99%

“…), where Λ (1) is the first coordinate of Λ and Λ (2) is the second coordinate. By the orthogonality of Λ, we have (Λ − Λ) \ {0} ⊂ Z( μ M,{D n } ).…”

Section: Introductionmentioning

confidence: 99%

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A Nonspectral Problem for Planar Moran–sierpinski Measures

Wei²

2023

Bull. Aust. Math. Soc.

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“…The conjecture was disproved in both directions on ℝ 𝑛 for 𝑛 ≥ 3, see [22,23,35]. Remarkably, both directions of the conjecture are still open in ℝ and ℝ 2 . Although the conjecture in its original form has been disproved, there is a clear connection between spectral sets and tilings, but the precise correspondence is still a mystery.…”

Section: Introductionmentioning

confidence: 99%

Fourier bases of the planar self‐affine measures with three digits

Chen

Liu

Yao

2023

Mathematische Nachrichten

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For an expansive real matrix 𝑀 =] and a noncollinear integer digit set 𝐷 = {(0, 0) 𝑡 , (𝛼 1 , 𝛼 2 ) 𝑡 , (𝛽 1 , 𝛽 2 ) 𝑡 } with 𝛼 2 − 2𝛽 2 ∉ 3ℤ, let 𝜇 𝑀,𝐷 be the self-affine measure defined by 𝜇 𝑀,𝐷 (⋅) = 1 3 ∑ 𝑑∈𝐷 𝜇 𝑀,𝐷 (𝑀(⋅) − 𝑑). In this paper, some necessary and sufficient conditions for 𝐿 2 (𝜇 𝑀,𝐷 ) contains an infinite orthogonal exponential set or 𝜇 𝑀,𝐷 to be a spectral measure are given.

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“…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.…”

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confidence: 99%

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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On self-similar spectral measures

Cited by 25 publications

References 31 publications

A Nonspectral Problem for Planar Moran–sierpinski Measures

A Nonspectral Problem for Planar Moran–sierpinski Measures

Fourier bases of the planar self‐affine measures with three digits

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

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