Reconstructions of signals of a known class from a given set of linear measurements

This paper investigates the error in reconstructions of a signal based on a given finite set of linear measurements, and presents two schemes that, if there is available a priori knowledge of the class of signals of which the measured signal is a member, can achieve a reduction of this error beyond the best that could be done without such knowledge. The error measure used is the supremum over the class of the
\mathcal{L}_2
distance between a signal and its reconstruction. The essence of the proposed reconstruction techniques is a coordinate transformation from the sampling subspace to a new reconstruction subspace known to be efficient for representation of signals of the given class. This study applies the theory of extremal subspaces and
n
widths of signal classes originated by Kolmogorov. Results are applied to the much studied class of time-concentrated band-limited signals. The measurement process is here assumed to be the convenient one of Nyquist rate time sampling. For this problem, plots of the error bounds and of several test functions and their reconstructions are presented, both for the proposed reconstructions, and for conventional cardinal-sampling-theorem reconstructions.