We explore the reconstruction of the gravitational lensing field of the
cosmic microwave background in real space showing that very little statistical
information is lost when estimators of short range on the celestial sphere are
used in place of the customary estimators in harmonic space, which are nonlocal
and in principle require a simultaneous analysis of the entire sky without any
cuts or excisions. Because virtually all the information relevant to lensing
reconstruction lies on angular scales close to the resolution scale of the sky
map, the gravitational lensing dilatation and shear fields (which unlike the
deflection field or lensing potential are directly related to the observations
in a local manner) may be reconstructed by means of quadratic combinations
involving only very closely separated pixels. Even though harmonic space
provides a more natural context for understanding lensing reconstruction
theoretically, the real space methods developed here have the virtue of being
faster to implement and are likely to prove useful for analyzing realistic maps
containing a galactic cut and possibly numerous small excisions to exclude
point sources that cannot be reliably subtracted.Comment: 21 pages, 8 figure
