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

**INTEGRAL OPERATORS BASIC**

IN RANDOM FIELDS ESTIMATION THEORY

IN RANDOM FIELDS ESTIMATION THEORY

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.

**Received: **April 18, 2005

**AMS Subject Classification: **35S15, 35R30, 45B05, 45P05, 62M09, 62M40

**Key Words and Phrases: **integral equations, pseudodifferential operators, random fields estimation, boundary-value problems, Fredholm operator

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2005

**Volume:** 20

**Issue:** 3