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We propose and analyze a low-complexity channel estimator for a multiuser multicarrier code division multiple access (MC-CDMA) downlink in a time-variant frequency-selective channel. MC-CDMA is based on orthogonal frequency division multiplexing (OFDM). The time-variant channel is estimated individually for every flat-fading subcarrier, assuming small intercarrier interference. The temporal variation of every subcarrier over the duration of a data block is upper bounded by the Doppler bandwidth determined by the maximum velocity of the users. Slepian showed that time-limited snapshots of bandlimited sequences span a low-dimensional subspace. This subspace is also spanned by discrete prolate spheroidal (DPS) sequences. We expand the time-variant subcarrier coefficients in terms of orthogonal DPS sequences we call Slepian basis expansion. This enables a time-variant channel description that avoids the frequency leakage effect of the Fourier basis expansion. The square bias of the Slepian basis expansion per subcarrier is three magnitudes smaller than the square bias of the Fourier basis expansion. We show simulation results for a fully loaded MC-CDMA downlink with classic linear minimum mean square error multiuser detection. The users are moving with 19.4 m/s. Using the Slepian basis expansion channel estimator and a pilot ratio of only 2%, we achieve a bit error rate performance as with perfect channel knowledge.