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Singular perturbation theory for a class of Fredholm integral equations arising in random fields estimation theory. (English) Zbl 1096.45003
The authors consider the equation $\varepsilon h(x,\varepsilon)+R h(x,\varepsilon)\equiv f(x)\, ,\qquad x\in D\subset\mathbb{R}^n\, ,\tag{1}$ where $$D$$ is a bounded domain with a smooth boundary $$\partial D$$ and $Rg(x)=\int_D R(x,y)g(y)\,dy\tag{2}$ is an integral term. The kernel $$R(x,y)$$ of (2) satisfies the equation
$Q(x,D_x)R(x,y)=P(x,D_x)\delta(x-y)\, ,\qquad R(x,y)=\mathcal{O}(1) \quad\text{ as}\quad | x-y| \to\infty\, ,$ where $$P(x,D_x)$$, $$Q(x,D_x)$$ are elliptic differential operators with smooth coefficients, while $$\delta(z)$$ is the Dirac’s delta function. Equation (1) with the limiting equation $$Rh=f$$ is basic in random fields estimation theory and the kernel $$R(x,y)$$ is non-negative definite $$\big(R\varphi,\varphi\big)\geq0$$ for all $$\varphi\in C^\infty_0(\mathbb{R}^n)$$, known as the covariance function. The asymptotic of the solution to equation (1) is constructed first in the case $$n=1$$ and then in the case $$n>1$$. Examples of applications are given.
##### MSC:
 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 60G35 Signal detection and filtering (aspects of stochastic processes) 60G60 Random fields 45M05 Asymptotics of solutions to integral equations
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