Spectrality of a class of self-affine measures on R2*The research is supported in part by the NNSF of China (Nos. 12071125, 11971500, 11831007 and 11771457), the Hunan Provincial NSF (No. 2019JJ20012). (23rd September 2021)

• Record Type: Journal Article
• Authors: Chen, Ming-Liang; Liu, Jing-Cheng; Wang, Xiang-Yang
• Is Part Of: Nonlinearity. Volume 34:Number 11(2021)
• Page Start: 7446
• Page End: 7469
• Publication Date: 2021-09-23
• Subjects: self-affine measures -- spectral measures -- spectra -- Hadamard triples; Primary 28A25 -- 28A80 -- Secondary 42C05 -- 46C05; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ac2493
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this work, we study the spectral property of a class of self-affine measures μ M, D on R 2 generated by an expanding real matrix M = diag ρ 1 − 1, ρ 2 − 1 and a non-collinear integer digit set D = { ( 0, 0 ) t, ( α 1, α 2 ) t, ( β 1, β 2 ) t } with α i − 2 β i ∉ 3 Z, i = 1, 2. We give the sufficient and necessary conditions so that μ M, D becomes a spectral measure, i.e., there exists a countable subset Λ ⊂ R 2 such that { e 2 πi ⟨ λ, x ⟩ : λ ∈ Λ} forms an orthonormal basis for L 2 ( μ M, D ). This extends the results of Dai, Fu and Yan (2021 Appl. Comput. Harmon. Anal. 52 63–81) and Deng and Lau (2015 J. Funct. Anal. 269 1310–1326).