Spectrality of Infinite Convolutions in $\mathbb{R}^d$

Jun Jie Miao, Zhiqiang Wang

Published 2022-10-16Version 1

In this paper, we study the spectrality of infinite convolutions in $\mathbb{R}^d$, where the spectrality means the corresponding square integrable function space admits a family of exponential functions as an orthonormal basis. Suppose that the infinite convolutions are generated by a sequence of admissible pairs in $\mathbb{R}^d$. First, we give two sufficient conditions for their spectrality by using the equi-positive condition and the integral periodic zero set of measures. Then, we apply these results to a class of infinite convolutions supported in $[0,1]^d$, and we obtain the spectrality under a natural assumption. Finally, we relax the assumption and get a weaker sufficient condition in the $d$-dimensional space, where $d\leq 3$.