Kotel'nikov-Shannon theorem for homogeneous on time isotropic random fields on the sphere and statistical simulation (Q2767630)

Let \(\xi(t,\theta_1,\ldots,\theta_{n-1},\varphi)\) be a real random field on \(R\times S_{n}\), where \((\theta_1,\ldots,\theta_{n-1},\varphi)\) are spherical coordinates, \(S_{n}\) is a unit sphere in \(R^{n}\). The authors prove that if \(\xi(t,u)\) is continuous in mean square isotropic on the sphere random field, then \(\xi(t,u)=\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}\xi_{m}^{l}(t) S_{m}^{l}(u)\), where \(\xi_{m}^{l}(t)\), \(m=0,1,\ldots\), \(l=1,2,\ldots,h(m,n)\), is a sequence of random processes such that \(M\xi_{m}^{l}(t)\xi_{m_1}^{l_1}(s)=\delta_{m}^{m_1}\delta_{l}^{l_1}b_{m}(t,s)\); \(b_{m}(t,s)\) is a sequence of positive definite kernels on \(R\times R\) such that \(\sum_{m=0}^{\infty}h(m,n)b_{m}(t,t)<\infty\); \(S_{m}^{l}(u)\), \(l=1,2,\ldots,h(m,n)\), are orthonormal spherical harmonics of order \(m\), \(h(m,n)=(2m+n-2)(m+n-3)!/((n-2)!m!)\). The Kotel'nikov-Shannon decomposition of \(\xi(t,\theta_1,\ldots,\theta_{n-1},\varphi)\) is obtained and an algorithm of statistical simulation for Gaussian random field, isotropic on the unit sphere, is proposed.