  This paper studies several aspects of signal reconstruction of sampled data
in spaces of bandlimited functions. In the first part, signal spaces are
characterized in which the classical sampling series uniformly converge, and we
investigate whether adaptive recovery algorithms can yield uniform convergence
in spaces where non-adaptive sampling series does not. In particular, it is
shown that the investigation of adaptive signal recovery algorithms needs
completely new analytic tools since the methods used for non-adaptive
reconstruction procedures, which are based on the celebrated Banach-Steinhaus
theorem, are not applicable in the adaptive case. The second part analyzes the
approximation of the output of stable linear time-invariant (LTI) systems based
on samples of the input signal, and where the input is assumed to belong to the
Paley-Wiener space of bandlimited functions with absolute integrable Fourier
transform. If the samples are acquired by point evaluations of the input signal
f, then there exist stable LTI systems H such that the approximation process
does not converge to the desired output H(f) even if the oversampling factor is
arbitrarily large. If one allows generalized measurements of the input signal,
then the output of every stable LTI system can be uniformly approximated in
terms of generalized measurements of the input signal. The last section studies
the situation where only the amplitudes of the signal samples are known. It is
shown that one can find specific measurement functionals such that signal
recovery of bandlimited signals from amplitude measurement is possible, with an
overall sampling rate of four times the Nyquist rate.
