The nonlinear space of signals allowing Wiener＇s generalized harmonic analysis (GHA), the linear bounded power signal spaces of Beurling, Marcinkiewicz, and Wiener, and a new linear bounded power space are studied from a control and systems theory perspective. Specifically, it is shown that the system power gain is given by the H-infinity norm of the system transfer function in each of these spaces for a large class of (power) stable finite and infinite dimensional systems. The GHA setup is shown to possess several limitations for the purpose of robustness analysis which motivates the use of the other more general (nonstationary) signal spaces. The natural double-sided time axis versions of bounded power signal spaces are shown to break the symmetry between Hardy space H-infinity methods and bounded power operators; e.g., the system transfer function being in H-infinity does not imply that a causal-linear time-invariant (LTI) system is bounded as an operator on any of the double-sided versions of the studied bounded power signal spaces.