A class of spectral Moran measures

Abstract

Let {Dk}k=1∞ be a sequence of digit sets in N and let {bk}k=1∞ be a sequence of integer numbers bigger than 1. We call the family {fk,Dk(x)=bk−1(x+d):d∈Dk,k⩾1} a Moran iterated function system (IFS), which is a natural generalization of an IFS. We prove, under certain conditions in terms of (bk,Dk), that the associated Moran measure μ is a spectral measure, i.e., there exists a countable set Λ⊂N such that {e2πiλx:λ∈Λ} is an orthonormal basis for L2(μ).