Smoothing methods comparison for CMB E- and B-mode separation
  The anisotropies of the B-mode polarization in the cosmic microwave
background radiation play a crucial role for the study of the very early
Universe. However, in the real observation, the mixture of the E-mode and
B-mode can be caused by the partial sky surveys, which must be separated before
applied to the cosmological explanation. The separation method developed by
Smith (\citealt{PhysRevD.74.083002}) has been widely adopted, where the edge of
the top-hat mask should be smoothed to avoid the numerical errors. In this
paper, we compare three different smoothing methods, and investigate the
leakage residuals of the E-B mixture. We find that, if the less information
loss is needed and the smaller region is smoothed in the analysis, the
\textit{sin}- and \textit{cos}-smoothing methods are better. However, if we
need a clean constructed B-mode map, the larger region around the mask edge
should be smoothed. In this case, the \textit{Gaussian}-smoothing method
becomes much better. In addition, we find that the leakage caused by the
numerical errors in the \textit{Gaussian}-smoothing method mostly concentrates
on two bands, which is quite easy to be reduced for the further E-B
separations.
