
HEALPix conventions
A bandlimited function f on the sphere can
be expanded in spherical harmonics, Y_{lm},
as
where denotes a unit vector pointing at polar angle
and
azimuth
. Here we have assumed that there is insignificant signal power in modes
with l>l_{max} and introduce the notation that all sums over m run from
l_{max} to l_{max} but all quantities with index lm vanish
for m>l. Our conventions for Y_{lm} are defined in subsection
A.4 below.
Pixelising corresponds to sampling it at
locations ,
. The sample
function values f_{p} can then be used
to estimate a_{lm}. A straightforward estimator is
where the superscript star denotes complex conjugation, and an equal weight was assumed for each pixel. This
zeroth order estimator, as well as higher order estimators, are implemented in the
Fortran90 facility anafast, included in the
package. Publications discussing these estimators and the generalised
quadrature problem on the sphere (Hivon et al. ) are in preparation.
Subsections
Eric Hivon
20100618

