Spectrality of planar self-affine measures with two-element digit set

Li, Jian-Lin; Zhou, Wen

doi:10.1007/s11425-011-4278-6

Cited by 15 publications

(8 citation statements)

References 20 publications

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“…The method of Section depends mainly upon the two‐element digit sets. There are a large class of the four‐element digit sets whose

μ_{M, D}

‐orthogonality and spectrality of

μ_{M, D}

can be obtained from the two‐element digit sets (see ). Except for the method of Section above, the following observation is also useful.…”

Section: A Methods From the Two‐element Digit Setsmentioning

confidence: 99%

“…□ Remark (i) The following example illustrates that the condition (a) or the condition (b) in Proposition cannot be omitted: Let

p_{1} \in (2 boldZ + 1) ∖ {\pm 1}, p_{2} \in (2 boldZ) ∖ {0}

and

b \in Z

. Consider the M and D given by

M = [\begin{matrix} right p_{1} & right b \\ right 0 & right p_{2} \end{matrix}] and D = \{((), \begin{matrix} 0 \\ 0 \end{matrix}), ((), \begin{matrix} 1 \\ 0 \end{matrix})\} .

Then, neither the condition (a) nor the condition (b) is satisfied; there exists no set

S \subset {boldZ}^{2}

such that

(() M^{- 1} D, S)

is a compatible pair;

μ_{M, D}

is a non‐spectral measure, and there are at most 2 mutually orthogonal exponential functions in

L^{2} (μ_{M, D})

(see). (ii) From (4.10), we know that

M_{1} : = M^{2}

and

D_{1} : = D + M D = D \oplus M D

are given by

M_{1} = [\begin{matrix} right p_{1}^{2} & right (p_{1} + p_{2}) b \\ right 0 & right p_{2}^{2} \end{matrix}] and D_{1} = \{((), \begin{matrix} 0 \\ 0 \end{matrix}),<...>\}

…”

Section: A Methods From the Two‐element Digit Setsmentioning

confidence: 99%

“…The condition det(M) ∈ 2Z corresponds to 6 cases (2.8). According to (5.9), these cases M α (α = 11,12,13,14,15,16) can be further divided into 96 cases M β,α (α = 11, 12, 13, 14, 15, 16; β = 1, 2, . .…”

Section: The Methods When Z Is Finitementioning

confidence: 99%

“…. , n), (4.9) we see that Then, neither the condition (a) nor the condition (b) is satisfied; there exists no set S ⊂ Z 2 such that M −1 D, S is a compatible pair; μ M,D is a non-spectral measure, and there are at most 2 mutually orthogonal exponential functions in L 2 (μ M,D ) (see [14]).…”

Section: Wwwmn-journalcommentioning

confidence: 99%

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Analysis of μ_{M, D}‐orthogonal exponentials for the planar four‐element digit sets

2013

Mathematische Nachrichten

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The self‐affine measure μM,D is a unique probability measure satisfying the self‐affine identity with equal weight. It only depends upon an expanding matrix M and a finite digit set D. In this paper we study the question of when the L2(μM,D)‐space has infinite families of orthogonal exponentials. Such research is necessary to further understanding the spectrality of μM,D. For a class of planar four‐element digit sets, we present several methods to deal with this question. The application of each method is also given, which extends the known results in a simple manner.

show abstract

“…The method of Section depends mainly upon the two‐element digit sets. There are a large class of the four‐element digit sets whose

μ_{M, D}

‐orthogonality and spectrality of

μ_{M, D}

can be obtained from the two‐element digit sets (see ). Except for the method of Section above, the following observation is also useful.…”

Section: A Methods From the Two‐element Digit Setsmentioning

confidence: 99%

“…□ Remark (i) The following example illustrates that the condition (a) or the condition (b) in Proposition cannot be omitted: Let

p_{1} \in (2 boldZ + 1) ∖ {\pm 1}, p_{2} \in (2 boldZ) ∖ {0}

and

b \in Z

. Consider the M and D given by

M = [\begin{matrix} right p_{1} & right b \\ right 0 & right p_{2} \end{matrix}] and D = \{((), \begin{matrix} 0 \\ 0 \end{matrix}), ((), \begin{matrix} 1 \\ 0 \end{matrix})\} .

Then, neither the condition (a) nor the condition (b) is satisfied; there exists no set

S \subset {boldZ}^{2}

such that

(() M^{- 1} D, S)

is a compatible pair;

μ_{M, D}

is a non‐spectral measure, and there are at most 2 mutually orthogonal exponential functions in

L^{2} (μ_{M, D})

(see). (ii) From (4.10), we know that

M_{1} : = M^{2}

and

D_{1} : = D + M D = D \oplus M D

are given by

M_{1} = [\begin{matrix} right p_{1}^{2} & right (p_{1} + p_{2}) b \\ right 0 & right p_{2}^{2} \end{matrix}] and D_{1} = \{((), \begin{matrix} 0 \\ 0 \end{matrix}),<...>\}

…”

Section: A Methods From the Two‐element Digit Setsmentioning

confidence: 99%

Section: The Methods When Z Is Finitementioning

confidence: 99%

Section: Wwwmn-journalcommentioning

confidence: 99%

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Analysis of μ_{M, D}‐orthogonal exponentials for the planar four‐element digit sets

2013

Mathematische Nachrichten

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show abstract

“…However, in the higher dimensions, there are only partial results concerning the spectrality of self-affine measures with two-element digit sets. Also, the above-mentioned conjecture in the introduction is still open in the case of two-element digit sets even in the plane R 2 (see [14]). The above Theorem 2.3 extends the corresponding result of [14] in a reasonable manner.…”

mentioning

confidence: 99%

A class of spectral self-affine measures with four-element digit sets

Yang

2015

Journal of Mathematical Analysis and Applications

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The self-affine measure μ M,D associated with an expanding matrix M ∈ M n (Z) and a finite digit set D ⊂ Z n is uniquely determined by the self-affine identity with equal weight. In this paper we construct a class of self-affine measures μ M,D with four-element digit sets in the higher dimensions (n ≥ 3) such that the Hilbert space L 2 (μ M,D ) possesses an orthogonal exponential basis. That is, μ M,D is spectral. Such a spectral measure cannot be obtained from the condition of compatible pair. This extends the corresponding result in the plane.

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A Class of Homogeneous Moran Spectral Measures with Eight-Element Digit Sets on $${\mathbb R}^4$$

Zhu

2021

Results Math

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Spectrality of planar self-affine measures with two-element digit set

Cited by 15 publications

References 20 publications

Analysis of μ_{M, D}‐orthogonal exponentials for the planar four‐element digit sets

Analysis of μ_{M, D}‐orthogonal exponentials for the planar four‐element digit sets

A class of spectral self-affine measures with four-element digit sets

A Class of Homogeneous Moran Spectral Measures with Eight-Element Digit Sets on $${\mathbb R}^4$$

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Spectrality of planar self-affine measures with two-element digit set

Cited by 15 publications

References 20 publications

Analysis of μM, D‐orthogonal exponentials for the planar four‐element digit sets

Analysis of μM, D‐orthogonal exponentials for the planar four‐element digit sets

A class of spectral self-affine measures with four-element digit sets

A Class of Homogeneous Moran Spectral Measures with Eight-Element Digit Sets on $${\mathbb R}^4$$

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Product

Resources

About

Analysis of μ_{M, D}‐orthogonal exponentials for the planar four‐element digit sets

Analysis of μ_{M, D}‐orthogonal exponentials for the planar four‐element digit sets