Singular integral equations and realization: A survey of the state space method

Abstract

Different methods for solving singular integral equations exist. One of the most recent methods is the so-called state space method. This method is based on the fact that a rational matrix function VV(^) which is analytic and invertible at infinity can be represented by vv(^): D * C(AI - A)-'B, (0.1) where A is a square matrix whose order may be larger than that of I,7()), and .8. C and D are matrices of appropriate sizes. The representation (0.1) allows one to reduce analytic problems about rational matrix functions to linear algebra ones involving constant matrices, and often it provides explicit and readily computable formulas for the solutions. In the last fifteen years the state space approach has proved to be effective in solving various problems of mathematical analysis (see the survey paper [BGK3]). In this mini-thesis we employ the state space method to solve singular integral equations. These equations serve as a tool to solve problems in numerous fields of application. For the general theory and examples of applications (see, for instance, [GKr], [M] and [V]).