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

Omarov, S. O.

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