IJPAM: Volume 62, No. 1 (2010)

COLLOCATION METHOD FOR SOLVING SOME INTEGRAL
EQUATIONS OF ESTIMATION THEORY

Alexander G. Ramm
Department of Mathematics
Kansas State University
Manhattan, KS 66506-2602, USA
e-mail: ramm@math.ksu.edu


Abstract.A class of integral equations $Rh=f$ basic in estimation theory is introduced. The description of the range of the operator $R$ is given. The operator $R$ is a positive rational function of a selfadjoint elliptic operator $\mathcal{L}$. This operator is defined in the whole space $\R^r$, it has a kernel $R(x,y)$, and $Rh:=\int_{D}R(x,y)h(y)dy$, where $D\subset \R^r$ is a bounded domain with a sufficiently smooth boundary $S$. Example of the equation of this type is $\int_{-1}^1 e^{-\vert x-y\vert}h(y)dy=f(x),\quad -1\leq x \leq 1.$ This equation has, in general, only distributional solutions. In estimation theory one is interested in the MOS (minimal order of singularity) solution to equation $Rh=f$. It is proved that such solution does exist and is unique for the class of equations defined by the author. A collocation method for numerical solution of equation $Rh=f$ in distributions is formulated and its convergence is proved.

Received: May 25, 2010

AMS Subject Classification: 162H12, 62M20, 62M40, 65R20, 45P05

Key Words and Phrases: estimation theory, integral equations, collocation method, distributional solution

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 62
Issue: 1