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

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.

### MSC:

 60G60 Random fields
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