UNIFIED HILBERT SPACE APPROACH TO ITERATIVE LEAST-SQUARES LINEAR SIGNAL RESTORATION.

Iterative least-squares solutions of the linear signal-restoration problem g equals Af are considered. First, several existing techniques for solving this problem with different underlying models are unified. Specifically, the following are shown to be special cases of a general iterative procedure for solving linear operator equations in Hilbert spaces: (1) a Van Cittert-type algorithm for deconvolution of discrete and continuous signals; (2) an iterative procedure for regularization when g is contaminated with noise; (3) a Papoulis-Gerchberg algorithm for extrapolation of continuous signals; (4) an iterative algorithm for discrete extrapolation of band-limited infinite-extent discrete signals right brace and the minimum-norm property of the extrapolation obtained by the iteration left brace ; and (5) a certain iterative procedure for extrapolation of band-limited periodic discrete signals. The Bialy algorithm also generalizes the Papoulis-Gerchberg iteration to cases in which the ideal low-pass operator is replaced by some other operators. In addition a suitable modification of this general iteration is shown. This technique leads us to new iterative algorithms for band-limited signal extrapolation. In numerical simulations some of these algorithms provide a fast reconstruction of the sought signal.