  Kepler seeks to detect sequences of transits of Earth-size exoplanets
orbiting Solar-like stars. Such transit signals are on the order of 100 ppm.
The high photometric precision demanded by Kepler requires detailed knowledge
of how the Kepler pixels respond to starlight during a nominal observation.
This information is provided by the Kepler pixel response function (PRF),
defined as the composite of Kepler's optical point spread function, integrated
spacecraft pointing jitter during a nominal cadence and other systematic
effects. To provide sub-pixel resolution, the PRF is represented as a
piecewise-continuous polynomial on a sub-pixel mesh. This continuous
representation allows the prediction of a star's flux value on any pixel given
the star's pixel position. The advantages and difficulties of this polynomial
representation are discussed, including characterization of spatial variation
in the PRF and the smoothing of discontinuities between sub-pixel polynomial
patches. On-orbit super-resolution measurements of the PRF across the Kepler
field of view are described. Two uses of the PRF are presented: the selection
of pixels for each star that maximizes the photometric signal to noise ratio
for that star, and PRF-fitted centroids which provide robust and accurate
stellar positions on the CCD, primarily used for attitude and plate scale
tracking. Good knowledge of the PRF has been a critical component for the
successful collection of high-precision photometry by Kepler.
