On spectral structure and spectral eigenvalue problems for a class of self similar spectral measure with product form

Abstract

Abstract Let μ be a Borel probability measure with compact support on R d . We say that μ is a spectral measure if there exists Λ ⊆ R d , called a spectrum of μ, such that E ( Λ ) ≔ { e − 2 π i ⟨ λ , x ⟩ } λ ∈ Λ forms an orthonormal basis for L 2(μ). In this paper, we study the structure of spectra for a class of self-similar spectral measure μ R,B with product form on R . We first give a partially characterize for E Λ to be a maximal orthogonal family in L 2(μ R,B ) by using the notion of maximal tree mapping. Based on this, we give a sufficient condition for a maximal orthogonal family E Λ (which corresponds to a maximal tree mapping) to be an orthonormal basis of L 2(μ R,B ). Moreover, we completely settle two types of spectral eigenvalue problems for μ R,B . Precisely, on the first case, for the model spectrum (simplest spectrum) of μ R,B , we characterize all possible real numbers t such that tΛ is also a spectrum of μ R,B . On the other case, we characterize all possible real numbers t such that there exists a countable set Λ such that Λ and tΛ are both spectra of μ R,B .