The existing low gain feedback, which is a parameterized family of stabilizing state feedback gains whose magnitudes approach zero as the parameter decreases to zero, has been designed in very specific ways. In this paper, by recognizing the l(infinity) and l(2) slow peaking phenomenon that exists in discrete-time systems under low gain feedback, more general notions of l(infinity) and l(2) norm vanishment are considered so as to provide a full characterization of the nonexistence of slow peaking phenomenon in some measured signals. Low gain feedback that does not lead to l(infinity) and l(2) slow peaking in the control input are respectively referred to as l(infinity) and l(2) low gain feedback. Based on the notions of l(infinity) and l(2) vanishment, not only can the existing low gain feedback be recognized as an l(infinity) low gain feedback, but also a new design approach referred to as the l(2) low gain feedback approach is developed for discrete-time linear systems. Parallel to the effectiveness of lee low gain feedback in magnitude constrained control, the l(2) low gain feedback is instrumental in the control of discrete-time systems with control energy constraints. The notions of l(infinity) and l(2)-vanishment also result in a systematic approach to the design of l(infinity) and l(2) low gain feedback by providing a family of solutions including those resulting from the existing design methods. (C) 2012 Elsevier Ltd. All rights reserved.