Fourier bases of a class of planar self-affine measures

2022/12/15 by Chen, Ming-Liang, Liu, Jing-Cheng, Wang, Zhi-Yong · 1 citation
#28A80 #46C05 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 28A25 #Secondary 42C05

paper · doi:10.48550/arxiv.2212.07577

Abstract

Let μ_M,D be the planar self-affine measure generated by an expansive integer matrix M∈ M₂(ℤ) and a non-collinear integer digit set D=\\beginpmatrix 0 0\endpmatrix,\beginpmatrix α₁ α₂ \endpmatrix, \beginpmatrix β₁ β₂ \endpmatrix, \beginpmatrix -α₁-β₁ -α₂-β₂ \endpmatrix\. In this paper, we show that μ_M,D is a spectral measure if and only if there exists a matrix Q∈ M₂(ℝ) such that (M,D) is admissible, where M=QMQ^-1 and D=QD. In particular, when α₁β₂-α₂β₁∉ 2\Bbb Z, μ_M,D is a spectral measure if and only if M∈ M₂(2ℤ).

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