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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 ${\mu }_{M,D}$ on ${\mathbb{R}}^{2}$ generated by an expanding integer matrix $M\in {M}_{2}\left(\mathbb{Z}\right)$ with $\mathrm{det}\left(M\right)\in 3\mathbb{Z}$ and a non-collinear integer digit set $D=\left\{{\left(0,0\right)}^{t},{\left({\alpha }_{1},{\alpha }_{2}\right)}^{t},{\left({\beta }_{1},{\beta }_{2}\right)}^{t}\right\}$ with ${\alpha }_{1}{\beta }_{2}-{\alpha }_{2}{\beta }_{1}\in 3\mathbb{Z}$. We give the sufficient and necessary conditions for ${\mu }_{M,D}$ to be a spectral measure, i.e., there exists a countable subset $\mathrm{\Lambda }\subset {\mathbb{R}}^{2}$ such that $E\left(\mathrm{\Lambda }\right)=\left\{{e}^{2\pi i〈\lambda ,x〉}:\lambda \in \mathrm{\Lambda }\right\}$ forms an orthonormal basis for ${L}^{2}\left({\mu }_{M,D}\right)$. This completely settles the spectrality of the self-affine measure ${\mu }_{M,D}$.