Reconstruction of signals from their linear mapping image

Many signal processing problems can be formulated as that of reconstructing a signal given an observed linear mapping (or transformation) of that signal. As an example, the extrapolation of a time-truncated version of a band-limited signal may be so characterized. More specifically, we shall herein consider the task of reconstructing a signal x, which lies in a prespecified closed subspace of a Hilbert space, and, in which the only information available is the linear mapping y = Lx of that signal. This reconstruction will be implemented by means of an algorithmic procedure. Necessary and sufficient conditions which ensure the algorithm´s convergence are presented.