Linear convolution integral equations with asymptotically almost periodic solutions

Abstract

Let μ \mu be a bounded Borel measure and f be asymptotically almost periodic. Conditions are found which ensure that certain bounded solutions of the linear convolution integral equation g ∗ μ = f g \ast \mu = f are asymptotically almost periodic. This result is also extended to the case where the measure μ \mu is replaced by a tempered distribution τ \tau for which convolution with bounded functions makes sense.