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Gaps in time series can produce spurious features in power spectrum estimates. These artifacts can be suppressed by averaging spectrum estimates obtained by first windowing the time series with a collection of orthogonal tapers. Such "multitaper" methods have been used for data without gaps since the early 1980s and for more general sampling schemes since the late 1980s. We introduce three families of tapers for time series with gaps. Two of the families solve optimization problems. They minimize bounds on different measures of bias. Computing them involves solving large eigenvalue problems with special structure that can be exploited to construct efficient algorithms. The third family solves no particular optimization problem but is inexpensive to compute and gives spectrum estimates that are quite similar to the other two for actual and simulated helioseismic data. All three families of gap-adapted multitaper estimates have lower variance and bias than the periodogram. In simulations of helioseismic data with gaps, standard methods for constructing confidence intervals for multitaper spectrum estimates, including parametric approximations and resampling in the temporal and spectral domains, all failed to attain their nominal confidence level. We present a novel resampling technique that, in the same simulations, gave confidence intervals that attained the correct confidence level.