On spectral structure and spectral eigenvalue problems for a class of self similar spectral measure with product form. (6th June 2022)

• Record Type: Journal Article
• Authors: Li, Jinjun; Wu, Zhiyi
• Is Part Of: Nonlinearity. Volume 35:Number 6(2022)
• Page Start: 3095
• Page End: 3117
• Publication Date: 2022-06-06
• Subjects: orthonormal basis of exponential -- spectral measures -- spectral structures -- spectral eigenvalue problems; 28A80 -- 42C05; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac6b0c
• Languages: English
• ISSNs: 0951-7715

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 t Λ 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 t Λ are both spectra of μ R, B .