Spectrality of a class of Moran measures with product-form digit sets

Abstract

In this paper, we investigate the spectrality of a class of Moran measures \(\mu _{\{R_{n}\},\{B_{n}\}}\) generated by a sequence of integers \(\{R_{n}\}_{n=1}^{\infty }\) and a sequence of product-form digit sets \(\{B_{n}\}_{n=1}^{\infty }\), where \(R_{n}=N^{q_{n}}\), \(B_{n}=\{0,1,\ldots ,N-1\}\oplus N^{p_{n}}\{0,1,\ldots ,N-1\}\) with \(N\ge 2\) and \(q_{n},~p_{n}\) are positive integers for \(n\ge 1\). We give some sufficient conditions for \(\mu _{\{R_{n}\},\{B_{n}\}}\) to be a spectral measure, i.e., there exists a countable set \(\Lambda \) such that \(\{e^{2\pi i\lambda \cdot x}:\lambda \in \Lambda \}\) is an orthonormal basis in \(L^2(\mu _{\{R_{n}\},\{B_{n}\}})\). Our results partially extend the work of Liu et al. [19], where \(p_n=p\) and \(q_n=q\) for all \(n\ge 1\). Furthermore, we show that our results can be extended to higher-dimensional settings.