Spectrality of a class of planar self-affine measures with three-element digit sets

Let μ M , D be the self-affine measure generated by an expanding integer matrix M ∈ M 2 ( Z ) and an integer three-element digit set D = { ( 0 , 0 ) T , ( α , β ) T , ( γ , η ) T } . In this paper, we show that if 3 ∣ det ( M ) and 3 ∤ α η - β γ , then L 2 ( μ M , D ) has an orthogonal basis of expo...

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Bibliographic Details
Published in:Archiv der Mathematik 2021-03, Vol.116 (3), p.327-334
Main Authors: Chen, Yan, Dong, Xin-Han, Zhang, Peng-Fei
Format: Article
Language:English
Subjects:
Exponential functions
Integers
Mathematics
Mathematics and Statistics
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Summary:Let μ M , D be the self-affine measure generated by an expanding integer matrix M ∈ M 2 ( Z ) and an integer three-element digit set D = { ( 0 , 0 ) T , ( α , β ) T , ( γ , η ) T } . In this paper, we show that if 3 ∣ det ( M ) and 3 ∤ α η - β γ , then L 2 ( μ M , D ) has an orthogonal basis of exponential functions if and only if M ∗ u ∈ 3 Z 2 , where u = ( η - 2 β , 2 α - γ ) T .
ISSN:0003-889X
1420-8938
DOI:10.1007/s00013-020-01554-0