We consider optimization problems associated to a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T-cycles of a differentiable function f : R → R of the form x(k + 1) = f(x(k)) + u(k) where u(k) = (a1−1)f(x(k))+a2f(x(k−T))+· · ·+aN f(x(k−(N −1)T)) , with a1 + · · · + aN = 1. Following an approach of Morgul, we associate to each periodic orbit of f, N ∈ N, and a1, . . . , aN an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. We prove that, given any 1- or 2-cycle of f, there exist N and a1, . . ., aN whose associated polynomial is Schur stable, and we find the minimal N that guarantees this stabilization. The techniques of proof will take advantage of extremal properties of the Fejer kernels found in classical harmonic analysis.