From top-hat masking to smooth transitions: P-filter and its application
  to polarized microwave sky maps
  In CMB science, the simplest idea to remove a contaminated sky region is to
multiply the sky map with a mask that is 0 for the contaminated region and 1
elsewhere, which is also called a top-hat masking. Although it is easy to use,
such top-hat masking is known to suffer from various leakage problems.
Therefore, we want to extend the top-hat masking to a series of semi-analytic
functions called the P-filters. Most importantly, the P-filters can seamlessly
realize the core idea of masking in CMB science, and, meanwhile, guarantee
continuity up to the first derivative everywhere. The P-filters can
significantly reduce many leakage problems without additional cost, including
the leakages due to low-, high-, and band-pass filtering, and the E-to-E,
B-to-B, B-to-E, and E-to-B leakages. The workings of the P-filter are
illustrated by using the WMAP and Planck polarization sky maps. By comparison
to the corresponding WMAP/Planck masks, we show that the P-filter performs much
better than top-hat masking, and meanwhile, has the potential to supersede the
principal idea of masking in CMB science. Compared to mask apodization, the
P-filter is ``outward'', that tends to make proper use of the region that was
marked as 0; whereas apodization is ``inward'', that always kills more signal
in the region marked as 1.
