KERNEL REPRESENTATION AND PROPERTIES OF DISCRETE-TIME INPUT-OUTPUT SYSTEMS

Discrete-time systems in a formal input-output setting are considered. Weak linearity, weak shift invariance, and weak nonanticipation are defined. The often overlooked fact that linear systems may not have a kernel representation is pointed out. Necessary and sufficient conditions for kernel representation on l(p) spaces are given. It is shown that a linear system can have infinitely many kernel representations and that properties such as nonanticipation, shift invariance, and boundedness need not be reflected in the structure of a kernel representation. It is argued that a system is logically distinct from a parametric representation of itself.