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We deal with iterative least-squares solutions of the linear signal-restoration problem g = Af. 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 BialyH. , Arch. Ration. Mech. Anal. 4, 166 (1959) 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 PapoulisA. , IEEE Trans. Circuits Syst. CAS-22, 735 (1975) ; GerchbergR. W. , Opt. Acta21, 709 (1974) ; (4) an iterative algorithm for discrete extrapolation of band-limited infinite-extent discrete signals and the minimum-norm property of the extrapolation obtained by the iteration JainA. RanganathS. , IEEE Trans. Acoust. Speech Signal Process. ASSP-29, (1981) ; and (5) a certain iterative procedure for extrapolation of band-limited periodic discrete signals TomV. , IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 1052 (1981). 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.
Sanz et al. (Tue,) studied this question.