Spectral structure of Moran Sierpinski-type measure on R2.

Let M n = diag [ 3 p n , 3 q n ] with p n , q n ⩾ 1 for all n ⩾ 1 and let D = { (0 , 0) t , (1 , 0) t , (0 , 1) t } . One can generate a Borel probability measure μ { M n } , D = δ M 1 − 1 D ∗ δ (M 2 M 1) − 1 D ∗ δ (M 3 M 2 M 1) − 1 D ∗ ⋯. Such measure μ { M n } , D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L 2 (μ { M n } , D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L 2 (μ { M n } , D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L 2 (μ { M n } , D) or not. [ABSTRACT FROM AUTHOR]