Interpolation problem for homogeneous and isotropic field. (Ukrainian, English) Zbl 1150.60028

Semenovs’ka, N.

Teor. Jmovirn. Mat. Stat. 74, 150-158 (2006); translation in Theory Probab. Math. Stat. 74, 171-179 (2007).

Let \(\xi(x)\), \(x\in \mathbb{R}^{n}\) be a homogeneous and isotropic random field, and let \((r,\bar\phi)=(r,\phi_1,\dots,\phi_{n-1})\) be a spherical coordinates of point \(x\in R^{n}\). The author proves that if \(X=\{x_1,\dots,x_{N}\}\) is a given set of observations on the sphere \(S_{n}\), \(x_{k}=(r,\bar\phi_{k})\), \(y=(\rho,\bar\psi)\), then the best interpolation of \(\xi(y)\) by \(\widehat{\xi(y)}\in H_{X}(r)=\text{Closure}\{\sum\alpha_{k}\xi(x_{k}),| x_{k}|=r\}\) in the sense of minimum of mean square error has a form \(\widehat{\xi(y)}=\sum_{k=1}^{N}\alpha_{k}\xi(x_{k})\), where \(\alpha=(\alpha_1,\dots,\alpha_{N})\) is a solution to the system \[ \sum_{k=1}^{N}\alpha_{k}\int_{0}^{\infty}{J_{(n-2)/2}(\lambda R_{kj})\over(\lambda R_{kj})^{(n-2)/2}}d\Phi(\lambda)=\int_{0}^{\infty}{J_{(n-2)/2}(\lambda R_{j})\over(\lambda R_{j})^{(n-2)/2}}d\Phi(\lambda), j=1,\dots,N, \] where \(R_{kj}=2r\sin(\theta_{kj}/2)\), \(R_{j}=2\sqrt{r^2+\rho^2-2r\rho\cos\theta_{j}}\), \(\theta_{kj}\) is the angle between vectors \(x_{k}\) and \(x_{j}\); \(\theta_{j}\) is the angle between vectors \(x_{j}\) and \(y\). The asymptotic behaviour of interpolation error is investigated.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60G60

Random fields

Keywords:

interpolation; homogeneous and isotropic random field; limit approximation error