  We generalize the technique of fringe-rate filtering, whereby visibilities
measured by a radio interferometer are re-weighted according to their temporal
variation. As the Earth rotates, radio sources traverse through an
interferometer's fringe pattern at rates that depend on their position on the
sky. Capitalizing on this geometric interpretation of fringe rates, we employ
time-domain convolution kernels to enact fringe-rate filters that sculpt the
effective primary beam of antennas in an interferometer. As we show, beam
sculpting through fringe-rate filtering can be used to optimize measurements
for a variety of applications, including mapmaking, minimizing polarization
leakage, suppressing instrumental systematics, and enhancing the sensitivity of
power-spectrum measurements. We show that fringe-rate filtering arises
naturally in minimum variance treatments of many of these problems, enabling
optimal visibility-based approaches to analyses of interferometric data that
avoid systematics potentially introduced by traditional approaches such as
imaging. Our techniques have recently been demonstrated in Ali et al. (2015),
where new upper limits were placed on the 21 cm power spectrum from
reionization, showcasing the ability of fringe-rate filtering to successfully
boost sensitivity and reduce the impact of systematics in deep observations.
