On the Spectrality of a Class of Moran Measures

Abstract

In this paper, we study the spectrality of a class of Moran measures $\mu_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n, D_n)\}^∞_{n=1},$ where $\mathcal{P} = \{p_n\}^∞_{n=1}$ is a sequence of positive integers with $p_n > 1$ and $\mathcal{D}= \{D_n\}^∞_n=1$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality #$D_n ∈ \{2, 3, N_n\}.$ We find a countable set $Λ ⊂ \mathbb{R}$ such that the set $\{e^{−2πiλx}|λ ∈ Λ\}$ is an orthonormal basis of $L^2 (\mu_{\mathcal{P},\mathcal{D}})$ under some conditions. As an application, we show that when $\mu_{\mathcal{P},\mathcal{D}}$ is absolutely continuous, $\mu_{\mathcal{P},\mathcal{D}}$ not only is a spectral measure, but also its support set tiles $\mathbb{R}$ with $\mathbb{Z}.$