2014/01/19 by Xin-Rong Dai, Xinrong Dai, Dai, Xinrong · 2 citations
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #Mathematical functions and polynomials #math.FA
paper · pdf · doi:10.48550/arxiv.1401.4630
openalex publication_date 2014/01/19 · arxiv created 2015/02/09 · arxiv updated 2015/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let μq, b be the Cantor measure associated with the iterated function system fi(x)=x/b+i/q, 0≤ i≤ q-1, where 2≤ q, b/q∈ \Z. In this paper, we consider spectra and maximal orthogonal sets of the Cantor measure μq, b and their rational rescaling. We introduce a quantity to measure level difference between a branch and its subbranch for the labeling tree corresponding to a maximal orthogonal set of the Cantor measure μq, b, and use certain boundedness property of that quantity as sufficient and necessary conditions for a maximal orthogonal set of the Cantor measure μq, b to be its spectrum. We show that the integrally rescaled set KΛ is still a spectrum if it is a maximal orthogonal set, and we provide a simple characterization for the integrally rescaled set to be a maximal orthogonal set. As an application of the above characterization, we find all integers K such that KΛ4 are spectra of the Cantor measure μ2, 4, where Λ4:=\∑n=0^∞ dn 4n: dn∈ \0, 1\\⊂ \Z is the first known spectrum for the Cantor measure μ2, 4. Finally we discuss rescaling spectra rationally and construct a spectrum Λ for the Cantor measure μq, b such that Λ/(b-1) is a maximal orthogonal set but not a spectrum.