Singular Perturbation Theory for a Class of Fredholm Integral Equations Arising in Random Fields Estimation Theory

Abstract.

A basic integral equation of random fields estimation theory by the criterion of minimum of variance of the estimation error is of the form Rh = f, where $Rh = \smallint\limits_D {R(x,y)h(y)dy,}$ and R(x, y) is a covariance function.

The singular perturbation problem we study consists of finding the asymptotic behavior of the solution to the equation $\varepsilon h(x,\varepsilon)+Rh(x,\varepsilon)=f(x),$ as $\varepsilon \to 0, \varepsilon > 0.$ The domain D can be an interval or a domain in Rⁿ, n > 1. The class of operators R is defined by the class of their kernels R(x,y) which solve the equation Q(x, D _x )R(x, y) = P(x, D _x )δ(x − y), where Q(x, D _x ) and Px, D _x ) are elliptic differential operators.