  We present an analytic theory for the resolution attainable via eclipse
mapping of exoplanets, based on the Fourier components of the brightness
distribution on the planetary disk. We find that the impact parameter
determines which features can and cannot be seen, via the angle of the stellar
edge relative to the axis of the orbit during the eclipse. We estimate the
signal-to-noise ratio as a function of mapping resolution, and use this to
determine the attainable resolution for a given star-planet system. We test
this theory against numerical simulations and find good agreement; in
particular, our predictions for the resolution as a function of stellar edge
angle are accurate to the simulated data to within 10% over a wide range of
angles. Our prediction for the number of spatial modes that can be constrained
given a light curve error is similarly accurate. Finally, we give a list of
exoplanets with the best expected resolution for observations with the NIRISS
SOSS, NIRSpec G395H, and MIRI LRS instruments on JWST.
