Structure of Signals and Systems with Non-Convergent Sampling Representation

Inhalt:
In this paper we study the approximation of stable linear time-invariant (LTI) systems by sampling series for signals in the Paley–Wiener space PW 1/π of bandlimited signals with absolutely integrable Fourier transform. It is known that there exist signals and systems such that the approximation process diverges unboundedly. We analyze the structure of the sets of signals and systems that create divergence and give a sufficient condition for the spaceability of the sets: If, for a given system, there exists a signal such that the approximation process diverges, then there exists a closed infinite dimensional subspace, all signals of which, except the zero signal, lead to divergence. We prove the same result for the set of systems with divergent approximation process.