Fredholm integral equations on the Euclidean motion group

Abstract

In this work, methods for the solution of Fredholm equations of the first kind with convolution kernel are presented, where all the functions in the integral equation are functions on the Euclidean motion group, and the convolution product is defined relative to the group operation. An application in which such equations arise is examined in detail. The properties of the Fourier transform of scalar-valued functions on the Euclidean motion group are reviewed and applied to an exactly solvable example. Standard regularization techniques are then adapted and illustrated for cases in which exact solutions are not possible.