Spaceability for Sets of Bandlimited Input Functions and Stable Linear Time-Invariant Systems with Divergence Behavior

The approximation of linear time-invariant systems by sampling series is studied for bandlimited input functions in the Paley-Wiener space PWπ. It has been known that there exist functions and systems such that the approximation process diverges. In this paper we identify a signal set and a system set with divergent Shannon sampling expression. We analyze the structure of these sets and prove that they are jointly spaceable, i.e., that each of them contains an infinite dimensional closed subspace, such that for any pair of function and system from these subspaces, except the zero elements, we have divergence.