2019/07/19 by Zheng-Yi Lu, Xin-Han Dong, Peng-Fei Zhang
paper · doi:10.1515/forum-2019-0062
Abstract Let μ M , D μM,D be a self-affine measure generated by an expanding diagonal matrix M ∈ M 3 ( ℝ ) M∈ M3(ℝ) with entries ρ 1 , ρ 2 , ρ 3 ρ1,ρ2,ρ3 and the digit set D = ( 0 , 0 , 0 ) t , ( 1 , 0 , 0 ) t , ( 0 , 1 , 0 ) t , ( 0 , 0 , 1 ) t D=\(0,0,0)t,(1,0,0)t,(0,1,0)t,(0,0,1)t\ . In this paper, we prove that for any ρ 1 , ρ 2 , ρ 3 ∈ ( 1 , ∞ ) ρ1,ρ2,ρ3∈(1,∞) , if ρ 1 , ρ 2 , ρ 3 ∈ ± x 1 r : x ∈ ℚ + , r ∈ ℤ + ρ1,ρ2,ρ3∈\± x(1)/(r):x∈ℚ+,r∈% ℤ+\ , then L 2 ( μ M , D ) L2(μM,D) contains an infinite orthogonal set of exponential functions if and only if there exist two numbers of ρ 1 , ρ 2 , ρ 3 ρ1,ρ2,ρ3 that are in the set ± ( p q ) 1 r : p ∈ 2 ℤ + , q ∈ 2 ℤ + - 1 and r ∈ ℤ + \±((p)/(q))(1)/(r):p∈ 2ℤ+,q∈ 2ℤ+-1% and r∈ℤ+\ . In particular, if ρ 1 , ρ 2 , ρ 3 ∈ p q : p , q ∈ 2 ℤ + 1 ρ1,ρ2,ρ3∈\(p)/(q):p,q∈ 2ℤ+1\ , then there exist at most 4 mutually orthogonal exponential functions in L 2 ( μ M , D ) L2(μM,D) , and the number 4 is the best possible.