The Spectrality of a Class of Fractal Measures on $\mathbb{R}^{n}$

Abstract

Let $M=\rho^{-1}I\in M_{n}(\mathbb{R})$ be an expanding matrix with 0 < ∣ ρ ∣ < 1 and $D\subset\mathbb{Z}^{n}$ be a finite digit set with 0 ∈ D and $\cal{Z}(m_{D})-\cal{Z}(m_{D})\subset\cal{Z}(m_{D})\cup\{0\}\subset m^{-1}\mathbb{Z}^{n}$ for a prime m, where $\cal{Z}(m_{D}):=\{x : \sum\nolimits_{d\in D}e^{2\pi \mathrm{i}\langle\lambda,x\rangle}=0\}$. Let μ_M,D be the associated self-similar measure defined by $\mu_{M,D}(\cdot)={1\over{\vert D\vert}}\sum\nolimits_{d\in D}\mu_{M,D}(M(\cdot)-d)$. In this paper, the necessary and sufficient conditions for L²(μ_M,D) to admit infinite orthogonal exponential functions are given. Moreover, by using the order theory of polynomial, we estimate the number of orthogonal exponential functions for all cases that L²(μ_M,D) does not admit infinite orthogonal exponential functions.