Spectrality of Infinite Convolutions and Random Convolutions

TL;DR

This paper investigates the spectral properties of infinite and random convolutions, characterizing when these measures admit orthonormal bases of exponential functions using Fourier transform zeros.

Contribution

It introduces a method to determine spectrality of infinite convolutions via Fourier zeros and characterizes spectrality in random convolutions for specific cases.

Findings

Almost all random convolutions are spectral measures.

Complete characterization of spectrality in certain cases.

Analysis of the integral periodic zeros set of Fourier transforms.

Abstract

In this paper, we explore spectral measures whose square integrable spaces admit a family of exponential functions as an orthonormal basis.Our approach involves utilizing the integral periodic zeros set of Fourier transform to characterize spectrality of infinite convolutions generated by a sequence of admissible pairs.Then we delve into the analysis of the integral periodic zeros set. Finally, we show that given finitely many admissible pairs, almost all random convolutions are spectral measures. Moreover, we give a complete characterization of spectrality of random convolutions in some special cases.