This paper presents a new Lagrangian heuristic to solve the general capacitated single item lot sizing problem (CSILSP) where the total costs of production, setup, and inventory are to be minimized. We introduce a pre-smoothing procedure to transform the problem into a CSILSP with non-customer specific time windows and relax constraints that are specific to the CSILSP. The resulting uncapacitated single item problems with non-customer specific production time windows can be solved in polynomial time. We also analyze the performance of the Lagrangian heuristic for solving the CSILSP with non-customer specific time windows. Finally, the heuristic is adapted to the customer specific case. The introduction of pre-smoothing and the relaxation of CSILSP-specific constraints help to decrease the gap between lower bounds and upper bounds from 26.22 to 1.39 %, on average. The heuristic can be used to solve aggregate production planning problems, or can be integrated into a general procedure to solve more complex lot sizing problems.