Spectrality of random convolutions generated by finitely many Hadamard triples

Abstract

Let {(Nj,Bj,Lj):1⩽j⩽m} be finitely many Hadamard triples in R . Given a sequence of positive integers {nk}k=1∞ and ω=(ωk)k=1∞∈{1,2,…,m}N , let μω,{nk} be the infinite convolution given by μω,nk=δNω1−n1Bω1∗δNω1−n1Nω2−n2Bω2∗⋯∗δNω1−n1Nω2−n2⋯Nωk−nkBωk∗⋯. In order to study the spectrality of μω,{nk} , we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if gcd(Bj−Bj)=1 for 1⩽j⩽m , then all infinite convolutions μω,{nk} are spectral measures. This implies that we may find a subset Λω,{nk}⊆R such that {eλ(x)=e2πiλx:λ∈Λω,{nk}} forms an orthonormal basis for L2(μω,{nk}) .