A sufficient and necessary condition for infinite orthogonal sets on some Moran measures

期刊论文
发表日期：

2024年10月15日

查看全文

S. Chen,
J.-C. Liu,
J. Su,
S. Wu

摘要

In this work we shall concentrate on fractal-harmonic analysis of a class of Moran measures. Let $\{M_n\}_{n=1}^{\infty}$ ${M_{n}}_{n = 1}^{\infty}$ be a sequence of expanding matrix in $M_2(\mathbb{Z})$ $M_{2} (Z)$ and $\{D_n\}_{n=1}^{\infty}$ ${D_{n}}_{n = 1}^{\infty}$ be a sequence of non-collinear integer digit sets satisfying $D_n= \left\{\begin{pmatrix}0\\0\end{pmatrix},\begin{pmatrix}\alpha_{n1}\\\alpha_{n2}\end{pmatrix},\begin{pmatrix}\beta_{n1}\\\beta_{n2}\end{pmatrix},\begin{pmatrix}-\alpha_{n1}-\beta_{n1}\\-\alpha_{n2}-\beta_{n2}\end{pmatrix} \right\}.$ $D_{n} = \{(\begin{matrix} 0 \\ 0 \end{matrix}), (\begin{matrix} α_{n 1} \\ α_{n 2} \end{matrix}), (\begin{matrix} β_{n 1} \\ β_{n 2} \end{matrix}), (\begin{matrix} - α_{n 1} - β_{n 1} \\ - α_{n 2} - β_{n 2} \end{matrix})\} .$ The associated Moran-type measure $\mu_{\{M_n\},\{D_n\}}$ $μ_{{M_{n}}, {D_{n}}}$ is generated by the infinite convolution $\mu_{\{M_n\},\{D_n\}}=\delta_{M_{1}^{-1}D_1}\ast\delta_{M_{1}^{-1}M_{2}^{-1}D_2}\ast\delta_{M_{1}^{-1}M_{2}^{-1} M_{3}^{-1}D_3}\ast\cdots$ $μ_{{M_{n}}, {D_{n}}} = δ_{M_{1}^{- 1} D_{1}} * δ_{M_{1}^{- 1} M_{2}^{- 1} D_{2}} * δ_{M_{1}^{- 1} M_{2}^{- 1} M_{3}^{- 1} D_{3}} * \dots$ in the weak $^*$ $^{*}$ -topology. Our result shows that if $\{\alpha_{n1}\alpha_{n2}\beta_{n1}\beta_{n2}\}_{n=1}^{\infty}$ ${α_{n 1} α_{n 2} β_{n 1} β_{n 2}}_{n = 1}^{\infty}$ is bounded, then $L^{2}(\mu_{\{M_n\},\{D_n\}})$ $L^{2} (μ_{{M_{n}}, {D_{n}}})$ admits an infinite orthogonal set of exponential functions if and only if there exists a subsequence $\{n_{k}\}_{k=1}^{\infty}$ ${n_{k}}_{k = 1}^{\infty}$ of $\{n_{k}\}_{k=1}^{\infty}$ ${n_{k}}_{k = 1}^{\infty}$ such that $\det(M_{n_{k}})\in 2\mathbb{Z}$ $det (M_{n_{k}}) \in 2 Z$ .