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In this paper, we develop and analyze the basic methodology for minimum-energy (ME) band-limited prediction of sampled time-variant flat-fading channels. This predictor is based on a subspace spanned by time-concentrated and band-limited sequences. The time-concentration of these sequences is matched to the length of the observation interval and the band-limitation is determined by the support of the Doppler power spectral density of the fading process. Slepian showed that discrete prolate spheroidal (DPS) sequences can be used to calculate the ME band-limited continuation of a finite sequence. We utilize this property to perform channel prediction. We generalize the concept of time-concentrated and band-limited sequences to a band-limiting region consisting of disjoint intervals. For a fading process with constant spectrum over its possibly discontiguous support we prove that the ME band-limited predictor is identical to a reduced-rank maximum-likelihood predictor which is a close approximation of a Wiener predictor. In current cellular communication systems the time-selective fading process is highly oversampled. The essential dimension of the subspace spanned by time-concentrated and band-limited sequences is in the order of two to five only. The prediction error mainly depends on the support of the Doppler spectrum. We exploit this fact to propose low-complexity time-variant flat-fading channel predictors using dynamically selected predefined subspaces. The subspace selection is based on a probabilistic bound on the reconstruction error. We compare the performance of the ME band-limited predictor with a predictor based on complex exponentials. For a prediction horizon of one eights of a wavelength the numerical simulation results show that the ME band-limited predictor with dynamic subspace selection performs better than, or similar to, a predictor based on complex exponentials with perfectly known frequencies. For a prediction horizons of three eights of a wave- length the performance of the ME band-limited predictor approaches that of a Wiener predictor with perfectly known Doppler bandwidth.