Spectrality of a class of self-affine measures on R2 *

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 2πi⟨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).