Spectral resolution of multidimensional homogeneous random fields that are isotropic in some of the variables. (English. Russian original) Zbl 0629.60056

Malyarenko, A. A.

Theory Probab. Math. Stat. 32, 69-75 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 66-72 (1985).

Let \(S(R^{m+n})\) be the Schwartz space of rapidly decreasing infinitely differentiable functions on \(R^{m+n}\), let E be a finite-dimensional Euclidean space. The author considers a mean-square-continuous generalized random field \(\xi\) (\(\phi)\), \(\phi \in S(R^{m+n})\), with values in E. The field \(\xi\) (\(\phi)\) is supposed to be homogeneous and isotropic in some of the variables. This means that \(\xi\) (\(\phi)\) is transforming under rotations in \(R^ n\) according to a given irreducible unitary representation of the group O(n) acting in E.
The paper presents a general form of the mean value and the correlation operator of \(\xi\) (\(\phi)\) and also a spectral resolution of the field in tems of stochastic integrals.

Reviewer: Yu.S.Mishura

MSC:

60G60	Random fields
60H05	Stochastic integrals
46F99	Distributions, generalized functions, distribution spaces

Keywords:

Schwartz space of rapidly decreasing infinitely differentiable functions; random field; homogeneous and isotropic; spectral resolution; stochastic integrals