Spectral structure and spectral eigenvalue problems of a class of self-similar spectral measures

Abstract

Let μ be a probability measure with compact support in R. The measure μ is called a spectral measure if there exists a countable set Λ⊆R, called a spectrum of μ, such that the family of exponential functions {e−2πiλx:λ∈Λ} forms an orthonormal basis for L2(μ). In this paper we study the structure of spectra and the real number t such that both Λ and tΛ are spectra for a class of self-similar spectral measures, which have symmetric spectra. Some examples are given to explain our theory.