A note on properties of BIBO stable linear discrete-space systems

Continuous linear shift-invariant operators that take bounded inputs into bounded outputs, with inputs and outputs real (or complex)-valued and defined on Z/sup m/ need not be convolutions. We show that all such operators have the property that they take S/sub 0/ into S/sub 0/ and also l/sub p/ into l/sub p/ for all real p/spl ges/1, where S/sub 0/ is the set of functions x that satisfies a certain natural generalization of the m=1 condition that x(/spl alpha/)/spl rarr/0 as |/spl alpha/|/spl rarr//spl infin/, and l/sub p/ is the family of pth-power summable functions defined on Z/sup m/. The proof makes use of a two-term representation for the operators.