  In this paper we describe the spherical harmonic transit telescope, a novel
formalism for the analysis of transit radio telescopes. This all-sky approach
bypasses the curved sky complications of traditional interferometry and so is
particularly well suited to the analysis of wide-field radio interferometers.
It enables compact and computationally efficient representations of the data
and its statistics that allow new ways of approaching important problems like
map-making and foreground removal. In particular, we show how it enables the
use of the Karhunen-Loeve transform as a highly effective foreground filter,
suppressing realistic foreground residuals for our fiducial example by at least
a factor twenty below the 21cm signal even in highly contaminated regions of
the sky. This is despite the presence of the angle-frequency mode mixing
inherent in real-world instruments with frequency-dependent beams. We show,
using Fisher forecasting, that foreground cleaning has little effect on power
spectrum constraints compared to hypothetical foreground-free measurements.
Beyond providing a natural real-world data analysis framework for 21cm
telescopes now under construction and future experiments, this formalism allows
accurate power spectrum forecasts to be made that include the interplay of
design constraints and realistic experimental systematics with twenty-first
century 21cm science.
