Theory and Methods Related to the Singular-Function Expansion and Landweber’s Iteration for Integral Equations of the First Kind

Abstract

For the Fredholm integral equation of the first kind, written notationally as $Kf = g$, $g \in L_2 [0,1]$, we study the behavior of the iteration \[ \hat f_k = \hat f_{k - 1} + DK^ * (g - K\hat f_{k - 1} ),\quad k = 1,2, \cdots ,\] both with respect to its convergence properties and its response to singular functions of K. Here $K^ * $ is the adjoint of K, $\hat f_0 $ is a suitable starting function and D is a fixed linear operator to be chosen. The general theory of singular functions is interpreted and extended for the study of the iteration, and a quantitative method of choosing D to shape the response to singular functions of K is derived. Several specific matrix and integral-equation examples are presented.