Spectrality and non-spectrality of a class of Moran measures

• Published in: Journal of mathematical analysis and applications Vol. 543; no. 2; p. 128937
• Main Author: Wang, Wei-Jie
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.03.2025
• Subjects: Fourier transform; Moran measure; Non-spectral measure; Orthogonal basis; Spectral measure; Fourier transform; Orthogonal basis; Non-spectral measure; Spectral measure; Moran measure
• ISSN: 0022-247X
• DOI: 10.1016/j.jmaa.2024.128937

Let {Mn}n=1∞∈M2(R) be a sequence of real expanding matrices and {Dn}n=1∞ be a sequence of digit sets withDn={(00),(anbn),(cndn)}, where |andn−bncn|=ϕ(n)∈Z, and letμ{Mn},{Dn}=δM1−1D1⁎δ(M1M2)−1D2⁎δ(M1M2M3)−1D3⁎⋯ be the associated Moran measure. This paper establishes a necessary and sufficient condition for μMn,Dn to be a spectral measure when each Mn is a diagonal matrix, and characterizes the structure of the spectrum Λ. Furthermore, we demonstrate that for any sequence of matrices {Mn}n=1∞ ∈ M2(Z) with det⁡(Mn)∉3Z, the space L2(μ{Mn},{Dn}) contains at most 9 mutually orthogonal exponential functions. The precise maximum cardinality of such functions is also determined.