2024/06/08 by Jun Jie Miao, Hongbo Zhao, Miao, Jun Jie +1
Mathematics · #42A85 #42C05 #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2406.05373
openalex publication_date 2024/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let μ denot the infinite convolution generated by \(Nk,Bk)\k=1^∞ given by μ=δ_N1-1B1∗δ(N1N2)-1B2∗…∗δ(N1N2⋯ Nk)-1Bk *⋯. where Bk is a complete residue system for each integer k>0. We write νgt;k=δ_Nk+1-1 Bk+1 * δ_(Nk+1 Nk+2)-1 Bk+2 * ⋯. Since the elements in Bk may have very large absolute values, the infinite convolution may not be compactly supported. In this paper, we study the necessary and sufficient conditions for such infinite convolutions being a spectral measure. Generally, for such infinite convolutions, the necessary conditions for spectrality mainly depend on the properties of the polynomials generated by the complete residue systems. The main result shows that if every Bk satisfies uniform discrete zero condition, and \ν>k\k=1^∞ is \it tight, then # Bk | Nk for all integers k≥ 2. For some special complete residue systems \Bk\k=1^∞, we provide the necessary and sufficient conditions for μ being a spectral measure.