In this paper, we study the spectrality of Sierpinski-Moran measure defined as an infinite convolution measure:

where \({\mathcal {D}}_n=\{(0,0)^t,(a_n,0)^t,(0,b_n)^t\}\subset {\mathbb {Z}}^{2}\) and \(\;M_n=\text {diag}(s_n,t_n)\in M_2(\mathbb Z)\) are \(2\times 2\) expanding diagonal matrices. Our goal is to investigate the existence of Fourier orthonormal basis for \(L^2(\mu _{\{M_n\},\{{\mathcal {D}}_n\}})\), i.e., find an exponential function system \(\{e^{2\pi i\langle \lambda ,x \rangle }\}_{\lambda \in \Lambda }\) forming an orthonormal basis for \(L^2(\mu _{\{M_n\},\{{\mathcal {D}}_n\}})\). Some sufficient conditions for this aim are given, and some spectra \(\Lambda \) are found.

在本文中，我们研究了定义为无限卷积测度的谢尔宾斯基-莫兰测度的频谱：

其中\({\mathcal {D}}_n=\{(0,0)^t,(a_n,0)^t,(0,b_n)^t\}\subset {\mathbb {Z}}^{ 2}\)和\(\;M_n=\text {diag}(s_n,t_n)\in M_2(\mathbb Z)\)是\(2\times 2\) 展开对角矩阵。我们的目标是研究\(L^2(\mu _{\{M_n\},\{{\mathcal {D}}_n\}})\)的傅立叶正交基的存在，即找到一个指数函数系统\(\{e^{2\pi i\langle \lambda ,x \rangle }\}_{\lambda \in \Lambda }\)形成\(L^2(\mu _{ \{M_n\},\{{\mathcal {D}}_n\}})\)。给出了该目标的一些充分条件，并找到了一些光谱\(\Lambda \)。