The Orthogonal Bases of Exponential Functions Based on Moran-Sierpinski Measures

Let A_n ∈ M₂ (ℤ) be integral matrices such that the infinite convolution of Dirac measures with equal weights

\(\mu_{\{A_{n},n\geq1\}}:=\delta_{A_{1}^{-1}\cal{D}}\ast\delta_{A_{1}^{-1}A_{2}^{-2}\cal{D}}\ast\cdots\) $${\mu }_{\{A_n,n\ge 1\}}:={\delta }_{A_1^{-1}\mathcal{D}}\ast {\delta }_{A_1^{-1}{A}_2^{-2}\mathcal{D}}\ast \cdots$$ is a probability measure with compact support, where \(\cal{D}=\{(0,0)^{t},(1,0)^{t},(0,1)^{t}\}\) $$\mathcal{D}=\{(0,0{)}^t,(1,0{)}^t,(0,1{)}^t\}$$ is the Sierpinski digit. We prove that there exists a set Λ ⊂ ℝ² such that the family {e^{2πi〈λ,x〉}: λ ∈ Λ} is an orthonormal basis of \(L^{2}(\mu_{\{A_{n},n\geq1\}})\) $$L^2({\mu }_{\{A_n,n\ge 1\}})$$ if and only if \({1\over{3}}(1,-1)A_{n}\in\mathbb{Z}^{2}\) $$\frac{1}{3}(1,-1)A_n\in {\mathbb{Z}}^2$$ for n ≥ 2 under some metric conditions on A_n.