ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60 Olenko, Andrew Yakovich, Kyiv University, Ukraine On some properties of correlation and spectral functions. Let ξ(x), x ∈ R^{n} be a real measurable square-mean continuous homogeneous isotropic random field with Eξ(x) = 0, Eξ^{2}(x) = 1, and the correlation function

B_{n}(r) = 2^{(n−2)/2}Γ (\frac{n}{2}) ∫_{0}^{∞} J_{\frac{n−2}{2}} (λr)(λr)^{(2−n)/2}dΦ(λ)

where J_{ν}(z) is the ν-order Bessel function of the first kind, Φ(λ) is a spectral function. Let f(λ) be a spectral density, which corresponds to Φ(λ).

1) Properties of positive definite spectral densities f(λ) are obtained.

2) Adjoint positive definite spectral densities f(λ) are studied.

3) Connections between spectral and correlation functions for different dimensions n are obtained.