## On the interpolation of random fields.(Russian)Zbl 0589.60042

Teor. Veroyatn. Mat. Stat. 31, 90-99 (1984).
Let $$\xi$$ (t), $$t\in R^ 1$$ be a separable random process with E $$\xi$$ (t)$$=0$$ and covariance function $B(t,s)=\int_{\Lambda}\int_{\Lambda}f(t,\lambda)\overline{f(s,\mu)}F(d\;lambda,d\mu),\quad \int_{\Lambda}\int_{\Lambda}F(d\lambda,d\mu)<\infty.$ If $$\sigma =\sup_{\lambda \in \Lambda} \overline{\lim}_{n\to \infty}| \partial^ nf(t,\lambda)/\partial t^ n|_{t=0}/^{1/n}<\infty$$, then for almost all sample functions $\xi (t)=\sum^{\infty}_{k=- \infty}[\sum^{N-1}_{m=1}\frac{\xi^{(m)}(a_ k)}{m!\quad}(t-a_ k)^ m]\times$
$\frac{\sin \alpha N^{-1}(t-a_ k)}{\alpha N^{- 1}(t-a_ k)}[\frac{\sin \beta N^{-1\quad}(t-a_ k)}{\beta N^{-1}(t- a_ k)}]^ NH_ q(t-a_ k)$ for fixed $$\alpha,\beta \leq \alpha N^{-1}-\sigma -q$$, where $$H_ q(z)$$ is an entire function of order $$q\leq \alpha N^{-1}-\beta$$, $$N\in Z_+$$, $$\alpha >0$$, $$\beta >0$$, $$\sigma <\alpha$$, $$\alpha N^{-1}-\beta >0$$. Some generalizations for random fields are considered.
Reviewer: N.Leonenko

### MSC:

 60G60 Random fields 60G10 Stationary stochastic processes

### Keywords:

interpolation; spectral representation