On self-similar spectral measures

Abstract

A probability measure μ on R is called a spectral measure if it has an exponential orthogonal basis for L2(μ). In this paper, we study the spectrality of the self-similar measure μρ,D generated by an iterated function system {τd(⋅)=ρ(⋅+d)}d∈D associated with a real number 0<ρ<1 and a finite set D⊂R. It can also be expressed asμρ,D=δρD⁎δρ2D⁎δρ3D⁎⋯=μk⁎μρ,D(ρ−k⋅), where μk is the convolutional product of the first k discrete measures. Until now, all known self-similar spectral measures are obtained from ρ−1∈N and spectral measures μk. We will show that these two conditions are also necessary under some natural assumptions. It improves significantly many results studied by recent research. As an application, we characterize a self-similar spectral measure associated with an integer tile.