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Spectrality of generalized Sierpinski-type self-affine measures
Applied and Computational Harmonic Analysis  (IF3.055),  Pub Date : 2021-05-11, DOI: 10.1016/j.acha.2021.05.001
Jing-Cheng Liu, Ying Zhang, Zhi-Yong Wang, Ming-Liang Chen

In this work, we study the spectral property of generalized Sierpinski-type self-affine measures μM,D on R2 generated by an expanding integer matrix MM2(Z) with det(M)3Z and a non-collinear integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t} with α1β2α2β13Z. We give the sufficient and necessary conditions for μM,D to be a spectral measure, i.e., there exists a countable subset ΛR2 such that E(Λ)={e2πiλ,x:λΛ} forms an orthonormal basis for L2(μM,D). This completely settles the spectrality of the self-affine measure μM,D.