Spectrality of a class of infinite convolutions

For a finite set D⊂Z and an integer b≥2, we say that (b,D) is compatible with C⊂Z if [e−2πidc/b]d∈D,c∈C is a Hadamard matrix. Let δE=1#E∑a∈Eδa denote the uniformly discrete probability measure on E. We prove that the class of infinite convolution (Moran measure) μb,{Dk}=δb−1D1⁎δb−2D2⁎⋯ is a spectral...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2015-10, Vol.283, p.362-376
Main Authors: An, Li-Xiang, He, Xing-Gang, Lau, Ka-Sing
Format: Article
Language:English
Subjects:
Admissible pairs
Compatible pairs
Convolution of measures
Moran measures
Self-similar measures
Spectral measures
Universally admissible
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Summary:For a finite set D⊂Z and an integer b≥2, we say that (b,D) is compatible with C⊂Z if [e−2πidc/b]d∈D,c∈C is a Hadamard matrix. Let δE=1#E∑a∈Eδa denote the uniformly discrete probability measure on E. We prove that the class of infinite convolution (Moran measure) μb,{Dk}=δb−1D1⁎δb−2D2⁎⋯ is a spectral measure provided that there is a common C⊂Z+ compatible to all the (b,Dk) and C+C⊆{0,1,…,b−1}. We also give some examples to illustrate the hypotheses and results, in particular, the last condition on C is essential.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2015.07.021