In this paper we analyze the behavior of system approximation processes for
stable linear time-invariant (LTI) systems and signals in the Paley-Wiener
space PW_\pi^1. We consider approximation processes, where the input signal is
not directly used to generate the system output, but instead a sequence of
numbers is used that is generated from the input signal by measurement
functionals. We consider classical sampling which corresponds to a pointwise
evaluation of the signal, as well as several more general measurement
functionals. We show that a stable system approximation is not possible for
pointwise sampling, because there exist signals and systems such that the
approximation process diverges. This remains true even with oversampling.
However, if more general measurement functionals are considered, a stable
approximation is possible if oversampling is used. Further, we show that
without oversampling we have divergence for a large class of practically
relevant measurement procedures.Comment: This paper will be published as part of the book "New Perspectives on
Approximation and Sampling Theory - Festschrift in honor of Paul Butzer's
85th birthday" in the Applied and Numerical Harmonic Analysis Series,
Birkhauser (Springer-Verlag). Parts of this work have been presented at the
IEEE International Conference on Acoustics, Speech, and Signal Processing
2014 (ICASSP 2014
