On moment-discretization and least-squares solutions of linear integral equations of the first kind

Abstract

Let K( s, t) be a continuous function on [0, 1] × [0, 1], and let K be the linear integral operator induced by the kernel K( s, t) on the space L 2[0, 1]. This note is concerned with moment-discretization of the problem of minimizing ‖ K x− y ‖ in the L 2-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝ 0 1 K( s i , t) x( t) dt = y( s i ), i = 1, 2, h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating K†y (where K † is the generalized inverse of K ), without recourse to the normal equation K∗Kx = K∗y, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.