Weak convergence and spectrality of infinite convolutions

By Wenxia Li, J. Miao, Zhiqiang Wang. Advances in Mathematics, 2022

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. Let { A k } ∞ k =1 be a sequence of finite subsets of R d satisfying that # A k ≥ 2 for all integers k ≥ 1. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolution where all sets A k ⊆ R d + and δ A = 1 # A P a ∈ A δ a . Then we study the spectrality of a class of infinite convolutions generated by Hadamard triples in R and construct a class of singular spectral measures without compact support. Finally we show that such measures are abundant, and the dimension of their supports has the intermediate-value property.

Subfields: Applied Mathematics, Mathematical Physics

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