Locally controllable regularization of Abel integral equations via operator complementation

The method of locally controllable regularization is described and analyzed for the model case of Abel's integral equation of the first kind on a bounded interval Ω. The perturbed data function is assumed to be given in L_q, the true solution is assumed to lie in a Lipschitz space H_p^λwith 1 ≤ p ≤ q ≤∞S, λ>0 (arbitrary but fixed). The error of the regularized approximate solution is measured in the L_p-norm corresponding to a subinterval of Ω. The approximation scheme is a finite-differencc imitation (of order determined by λ) to Abel's classical solution formula.