Weak convergence and spectrality of infinite convolutions

作者：Li, Wenxia; Miao, Jun Jie; Wang, Zhiqiang^*

来源：Advances in Mathematics, 2022, 404: 108425.

DOI：10.1016/j.aim.2022.108425

• Weak convergence
• Infinite convolution
• Spectral measure

Let {Ak}(k =1)(infinity) 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 & nbsp;nu = delta(A1) * delta(A2) * .& nbsp; .& nbsp; .& nbsp;* delta(An) * .& nbsp;.& nbsp;.& nbsp; E where all sets A(k) subset of & nbsp;& nbsp;R-+(d) and delta(A) = 1/# A( )sigma(a is an element of A)delta(a). Then #A 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.