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 L 2 ( μ ) . In this paper, we study the spectrality of the self-similar measure μ ρ , D generated by an iterated function system { τ d ( ⋅ ) = ρ ( ⋅ + d ) } d ∈ D associated with a real number 0 ρ 1 and a finite set D ⊂ R . It can also be expressed as μ ρ , D = δ ρ D ⁎ δ ρ 2 D ⁎ δ ρ 3 D ⁎ ⋯ = μ k ⁎ μ ρ , D ( ρ − k ⋅ ) , where μ k 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 μ k . 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.