On the bandlimited extrapolation of discrete signals

The extrapolation of a bandlimited signal from observations taken over a finite interval is an important problem in applications such as spectral estimation and image processing. The extrapolation of bandlimited analog signals is fundamentally different, however, from the band-limited extrapolation of discrete time sequences. Specifically, as has been previously noted, a unique solution to the discrete problem does not exist. In this paper, this fact is demonstrated in a new and convincing way. In particular, two methods are presented for constructing a sequence which, in the frequency domain, is band-limited to an arbitrary cutoff frequency and which, in the time domain, is equal to zero over an interval of arbitrary length. The importance of the existence of these multiple solutions to the general extrapolation problem is emphasized and questions are raised regarding the need for additional constraints in the discrete band-limited extrapolation problem.