On Asymptotic Behavior at Infinity and the Finite Section Method for Integral Equations on the Half-line

We consider integral equations on the halfline of the form x(s) − ∫∞ 0 k(s, t)x(t) dt = y(s) and the finite section approximation xβ to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e., xβ exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k+h. The class of perturbations allowed includes all compact and some noncompact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which (1 + s)px(s) is bounded. With the additional assumption that |k(s, t)| ≤ |κ(s − t)|, where κ ∈ L1(R) and κ(s) = O(s−q) as s → +∞, for some q > 1, we show that the finite-section method is stable in the weighted space for 0 ≤ p ≤ q, provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x − xβ and precise information on the asymptotic behavior at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a WienerHopf integral equation arising in acoustics.