Theory and Methods Related to the Singular-Function Expansion and Landweber's Iteration for Integral Equations of the First Kind - Strand, Otto Neall (SIAM journal on numerical analysis, 1974)
For the Fredholm integral equation of the first kind, written notationally as Kf = g, g ∈ L2 [ 0, 1 ], we study the behavior of the iteration $\hat{f}_k = \hat{f}_{k - 1} + DK^*(g - K \hat{f}_{k - 1}),\quad k = 1, 2, …,$ both with respect to its convergence properties and its response to singular functions of K. Here K* is the adjoint of K, f̂0 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.