Spaceability for sets of bandlimited input functions and stable linear time-invariant systems with finite time blowup behavior

The approximation of linear time-invariant systems by sampling series is studied for bandlimited input functions in the Paley–Wiener space $\mathcal{PW}_\pi^1$, i.e., bandlimited signals with absolutely integrable Fourier transform. 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 divergence, i.e., a finite time blowup of the Shannon sampling expression. We analyze the structure of these sets and prove that they are jointly spaceable, i.e., each of them contains an infinite-dimensional closed subspace such that for any function and system pair from these subspaces, except for the zero elements, we have divergence.