Let G be a locally compact abelian group and let M (G) be the measure algebra of G. Assume that mu is an element of M (G) is power bounded, that is, sup(n >= 0) parallel to mu(n)parallel to(1) < infinity. This paper is concerned mainly with finding necessary and sufficient conditions for strong convergence of iterates of the convolution operators T(mu)f : = mu * f in L-P (G) (1 <= p < infinity) spaces. Some related problems are also discussed. (C) 2019 Elsevier Masson SAS. All rights reserved.