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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.
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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