Collocation method for solving some integral equations of estimation theory

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 L. This operator is defined in the whole space R, it has a kernel R(x, y), and Rh := R D R(x, y)h(y)dy, where D ⊂ R is a bounded domain with a sufficiently smooth boundary S. Example of the equation of this type is R 1 −1 e −|x−y|h(y)dy = f(x), −1 ≤ x ≤ 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. MSC: 162H12, 62M20, 62M40, 65R20, 45P05