Spectrality of a class of Moran measures

Abstract

Let $\mu$ be a Borel probability measure on ${\mathbb {R}}^n$. We call $\mu$ a spectral measure if there exists a countable set $\Lambda \subset {\mathbb {R}}^n$ such that $E_\Lambda :=\{e^{2\pi i<\lambda ,x>}:\lambda \in \Lambda \}$ forms an orthogonal basis for the Hilbert space $L^2(\mu )$. Let the measure $\mu _{\{M,{\mathcal {D}}_n\}}$ be defined by the following expression $\mu _{\{M,{\mathcal {D}}_n\}}=\delta _{M^{-1}{\mathcal {D}}_1}*\delta _{M^{-2}{\mathcal {D}}_2}*\cdots$, where $M=\text {diag}(\rho ^{-1},\rho ^{-1})$ with $|\rho |<1$, and ${\mathcal {D}}_n=\left\{ (0,0)^t,(a_n,b_n)^t,(c_n,d_n)^t)\right\}$ with $|a_n d_n-b_n c_n|=1$ for all $n\ge 1$. This paper focuses on the spectrality of a class of Moran measures $\mu _{\{M,{\mathcal {D}}_n\}}$ in ${\mathbb {R}}^2$. We give some sufficient conditions for $\mu _{\{M,{\mathcal {D}}_n\}}$ to be a spectral measure. Also some necessary conditions are obtained. Moreover, the spectrality of the above Moran measures are also used to determine the spectrality of certain self-affine measures.