Weighted Paley–Wiener Theorem, with applications to stability of the linear multi-step methods for Volterra equations in Hilbert spaces

We study the weighted l 1 Paley–Wiener Theorem, with applications to stability of the linear multi-step time discretization for a linear Volterra equation of scalar type in a Hilbert space. The results extend to weighted l 1 stability some recent l 1 stability theorem due to the author (Numer. Math. 109 (2008) 571–595). Our theoretical results are applied to investigate the asymptotic behavior as t n → ∞ of the numerical solutions of certain integro-differential equations in Hilbert space.