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

TL;DR

This paper investigates the conditions under which infinite convolutions in Euclidean space have a spectral structure, meaning their associated function spaces can be spanned by exponential orthonormal bases, with new criteria provided.

Contribution

The paper introduces two new sufficient conditions for spectrality of infinite convolutions in $\

Findings

Established conditions based on equi-positivity and zero sets of Fourier transforms.

Proved spectrality for specific classes of infinite convolutions.

Provided a framework for analyzing spectrality in higher-dimensional spaces.

Abstract

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 $\mathbb{R}^d$. We give two sufficient conditions for their spectrality by using the equi-positivity condition and the integral periodic zero set of Fourier transform. By applying these results, we show the spectrality of some specific infinite convolutions in $\mathbb{R}^d$.