Finite length band-limited extrapolation of discrete signals

The problem of finite-length extrapolation of bandlimited sequences is formulated in terms of an energy minimization problem. Given a finite subset of samples (not necessarily consecutive) and a band limit ω_s, it is shown how to find an extrapolated sequence that agrees with the given subset of samples and has the minimum energy within the band ω_s<|ω|<π. The solution involves the inversion of a symmetric positive definite Toeplitz matrix. It is known that a bandlimited signal can be compressed and reconstructed from its nonuniform subsamples as long as the overall sampling rate is equal to or above the Nyquist rate of the signal. The above minimum energy method can also be used for reconstructing bandlimited signals from their subsamples. Upper and lower bounds on the mean square error or reconstruction are found to be related to the eigenvalues of the Toeplitz matrix and the out-of-band energy of the original sequence