A pixelated signal *f*(*p*) is the average within each pixel *p* (with surface
area
) of the underlying signal

f(p) |
(29) |

where

f(p) |
(30) |

where

is the Spherical Harmonic Transform of the pixel

However, complete analysis of a pixelated map with the exact *w*_{lm}(*p*)
defined above would be computationally intractable (because of azimutal
variation of pixel shape over the polar caps of the **HEALPix** grid),
and some simplifying asumptions have to be
made. If the pixel is small compared to the signal correlation length
(determined by the beam size), the exact structure of the pixel can be ignored
in the subsequent analysis and we can *assume*

w_{lm}(p) = w_{l}(p) Y_{lm}(p) |
(32) |

where we introduced the

which is independent of the pixel location on the sky.

If we assume all the pixels to be identical, the power spectrum of the
pixelated map,
, is related to the hypothetical unpixelated
one,
, by

where the effective pixel window function

This function is provided with the

The pixel window functions are now available for both temperature and polarization.

For
, those window functions are computed exactly using
Eqs. (33) and (35). For
the
calculations are too costly to be done exactly at all *l*. The temperature
windows are
extrapolated from the case
assuming a scaling in *l* similar
to the one exhibited by the window of a tophat pixel. The polarization
windows are assumed to be proportional to those for temperature, with a
proportionality factor given by the exact calculation of *w*_{l} at low
*l*.

Because of a change of the extrapolation scheme used, the temperature window
functions provided with **HEALPix** 1.2 and higher for
are slighty different from those
provided with **HEALPix** 1.1. For a given , the relative difference
increases almost linearly with *l*, and is of the order of
at
and
at
.

Version 3.31, 2017-01-06