# IJPAM: Volume 20, No. 3 (2005)

INTEGRAL OPERATORS BASIC
IN RANDOM FIELDS ESTIMATION THEORY

Alexander Kozhevnikov, Alexander G. Ramm
Department of Mathematics
University of Haifa
Mount Carmel, Haifa, 31905, ISRAEL
e-mail: kogevn@math.haifa.ac.il
Department of Mathematics
Kansas State University
Manhattan, KS 66506-2602, USA
e-mail: ramm@math.ksu.edu
url: https://www.math.ksu.edu/ramm

Abstract.The paper deals with the basic integral equation of random field estimation theory by the criterion of minimum of variance of the error estimate. This integral equation is of the first kind. The corresponding integral operator over a bounded domain in is weakly singular. This operator is an isomorphism between appropriate Sobolev spaces. This is proved by a reduction of the integral equation to an elliptic boundary value problem in the domain exterior to . Extra difficulties arise due to the fact that the exterior boundary value problem should be solved in the Sobolev spaces of negative order.