Weak convergence and spectrality of infinite convolutions

Wenxia Li; Jun Jie Miao; Zhiqiang Wang

doi:10.1016/j.aim.2022.108425

Weak convergence and spectrality of infinite convolutions

Wenxia Li
, Jun Jie Miao
, Zhiqiang Wang^*

^*Corresponding author for this work

School of Mathematical Sciences

East China Normal University

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

Abstract

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 ν=δ_A₁⁎δ_A₂⁎⋯⁎δ_{A_n}⁎⋯, where all sets A_k⊆R₊^d and [Formula presented]. 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.

Original language	English
Article number	108425
Journal	Advances in Mathematics
Volume	404
DOIs	https://doi.org/10.1016/j.aim.2022.108425
State	Published - 6 Aug 2022

Keywords

Infinite convolution
Spectral measure
Weak convergence

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10.1016/j.aim.2022.108425

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title = "Weak convergence and spectrality of infinite convolutions",

abstract = "Let \{Ak\}k=1∞ be a sequence of finite subsets of Rd satisfying that \#Ak≥2 for all integers k≥1. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolution ν=δA1⁎δA2⁎⋯⁎δAn⁎⋯, where all sets Ak⊆R+d and [Formula presented]. 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.",

keywords = "Infinite convolution, Spectral measure, Weak convergence",

author = "Wenxia Li and Miao, \{Jun Jie\} and Zhiqiang Wang",

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year = "2022",

month = aug,

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T1 - Weak convergence and spectrality of infinite convolutions

AU - Li, Wenxia

AU - Miao, Jun Jie

AU - Wang, Zhiqiang

PY - 2022/8/6

Y1 - 2022/8/6

N2 - Let {Ak}k=1∞ be a sequence of finite subsets of Rd satisfying that #Ak≥2 for all integers k≥1. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolution ν=δA1⁎δA2⁎⋯⁎δAn⁎⋯, where all sets Ak⊆R+d and [Formula presented]. 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.

AB - Let {Ak}k=1∞ be a sequence of finite subsets of Rd satisfying that #Ak≥2 for all integers k≥1. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolution ν=δA1⁎δA2⁎⋯⁎δAn⁎⋯, where all sets Ak⊆R+d and [Formula presented]. 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.

KW - Infinite convolution

KW - Spectral measure

KW - Weak convergence

UR - https://www.scopus.com/pages/publications/85129321252

U2 - 10.1016/j.aim.2022.108425

DO - 10.1016/j.aim.2022.108425

M3 - 文章

AN - SCOPUS:85129321252

SN - 0001-8708

VL - 404

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108425

ER -

Weak convergence and spectrality of infinite convolutions

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