On maximizing linear system output energy with an input envelope constraint

This correspondence addresses the problem of finding an input signal u that maximizes the output energy of a linear filter subject to the constraint |u|⩽e where e is a specified envelope. The main contribution is to provide a new direct proof of the necessity of a certain fixed point condition. Using the weak* topology on the input signal space, the author also proves certain compactness and continuity properties that further illuminate the nature of the problem