We analyze out-of-band (OOB) emissions in the massive multi-user (MU) multiple-input multiple-output (MIMO) downlink. We focus on systems in which the base station (BS) is equipped with low-resolution digital-to-analog converters (DACs) and orthogonal frequency-division multiplexing (OFDM) is used to communicate to the user equipments (UEs) over frequency-selective channels. We demonstrate that analog filtering in combination with simple frequency-domain digital predistortion (DPD) at the BS enables a significant reduction of OOB emissions, but degrades the signal-to-interference-noise-and-distortion ratio (SINDR) at the UEs and increases the peak-to-average power ratio (PAR) at the BS. We use Bussgang's theorem to characterize the tradeoffs between OOB emissions, SINDR, and PAR, and to study the impact of analog filters and DPD on the error-rate performance of the massive MU-MIMO-OFDM downlink. Our results show that by carefully tuning the parameters of the analog filters, one can achieve a significant reduction in OOB emissions with only a moderate degradation of error-rate performance and PAR.