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Extensions of Laba‐Wang's condition for spectral pairs
Author(s) -
Li JianLin
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400070
Subject(s) - mathematics , spectral measure , affine transformation , dimension (graph theory) , measure (data warehouse) , combinatorics , pure mathematics , matrix (chemical analysis) , discrete mathematics , database , computer science , materials science , composite material
Let μ M , Dbe a self‐affine measure associated with an expanding matrix M ∈ M n ( Z )and a finite digit set D ⊂ Z n . We consider in this paper the spectrality of μ M , D . In the case when ( M − 1 D , S ) is a compatible pair for some S ⊂ Z n , a necessary condition is obtained for the spectral pair ( μ M , D , Λ ( M , S ) ) . This condition is shown to be equivalent to the known necessary conditions for the same spectral pair. Moreover, we prove that all these necessary conditions are not sufficient in the higher dimensions, but they are sufficient in the dimension one. This extends Laba‐Wang's condition for spectral pairs.