On the renewal equation
Samuel Karlin · Pacific Journal of Mathematics · 1955
Σ, M l I _oo un exist. Similar results are valid for the circumstance where does not exist. Then we deal only with the case where an ~ 0 for n . t >_ 0 a strongly continuous semi-group of operators acting either on the space of bounded functions (M) or integrable functions ( L ) with | | Γ ( ί ) . | | < 1. Let A denote the infinitesimal generator of Tit) and let df it) define a nonlatt ice distribution with finite first moment on [0, col. If u belongs to iU) we consider u ( t ) (°° T ( t ) d f ( t ) ] u = v L J o J where v belongs to ( L ). The linear operator ί°° Tit)dfit) Jo is well defined either over iM) or ( L ) into itself. Put rit) = 1 f it), then r G L and the Fourier transform of r never vanishes . Since r is monotonic decreasing and in L it can be easi ly shown that