Let the infinite convolutions \begin{ equation*} \mu_{ \{ R_{ k}\},\{ D_{ k}\}}=\delta_{ R_{ 1}^{ -1}D_{ 1}}*\delta_{ R_{ 1}^{ -1}R_{ 2}^{ -1}D_{ 2}}*\delta_{ R_{ 1}^{ -1}R_{ 2}^{ -1}R_{ 3}^{ -1}D_{ 3}}*\dotsi \end{ equation*} be generated by the sequence of pairs \(\{ (R_k,D_k) \}_{ k=1}^{ \infty} \), where $ R_k\in M_n(\mathbb{ Z})$ is an expanding integer matric, $D_k$ is a finite integer digit sets that satisfies the following two conditions: (i). \( \# D_k = m \) and \( m>2 \) is a prime; (ii). \( \{ x: \sum_{ d\in D_{ k}}e^{ 2\pi i\langle d,x \rangle}=0\} =\cup_{ i=1}^{ \phi(k)}\cup_{ j=1}^{ m-1}(\frac{ j}{ m}\nu_{ k,i}+\mathbb{ Z}^{ n}) \) for some \( \nu_{ k,i} \in \{ (l_1, \cdots, l_n)^t : l_i \in [1, m-1] \cap \mathbb{ Z}, 1\leq i\leq n \} \). In this paper, we study the spectrality of $\mu_{ \{ R_{ k}\},\{ D_{ k}\}}$, and some necessary and sufficient conditions for \( L^{ 2}(\mu_{ \{ R_{ k}\},\{ D_{ k}\}}) \) to have an orthogonal exponential function basis are established. Finally, we discuss the explanations and applications of our results.
Categories: math.CA math.DS