A Banach Space Property for Signal Spaces With Applications for Sampling and System Approximation

The Banach-Steinhaus theorem, one of the fundamental results in functional analysis, completely characterizes the convergence of linear approximation processes. If the condition of boundedness is violated, then the principle of uniform boundedness implies the unbounded divergence of the approximation process on a residual set. In this paper we give a sufficient condition for Banach spaces that guarantees the unbounded divergence not only for a residual set but rather for a set that contains an infinite dimensional closed subspace after the zero element has been added. We further show that many important signal space, e.g., Paley-Wiener and Bernstein spaces, possess this property, and demonstrate consequences for the convergence behavior of sampling series and system approximation processes.