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

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 \[ f_k = fk - 1 + DK^ * (g - K fk - 1 ), k = 1,2, ⋯ ,\] both with respect to its convergence properties and its response to singular functions of K. Here K^ * is the adjoint of K, f₀ 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.