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HEALPix conventions

A bandlimited function f on the sphere can be expanded in spherical harmonics, Ylm, as
$\displaystyle f({ \gamma})$ $\textstyle \myequal $ $\displaystyle \sum_{l=0}^{l_{max}}\sum_{m}a_{lm}Y_{lm}(\gamma),$ (1)

where ${{\gamma}}$ denotes a unit vector pointing at polar angle $\theta\in[0,\pi]$ and azimuth $\phi\in[0,2\pi)$. Here we have assumed that there is insignificant signal power in modes with l>lmax and introduce the notation that all sums over m run from -lmax to lmax but all quantities with index lm vanish for m>l. Our conventions for Ylm are defined in subsection A.4 below.

Pixelising $f({ \gamma})$ corresponds to sampling it at $N_{\rm {pix}}$ locations $\gamma_{p}$, $p\in[0,N_{\rm {pix}}-1]$. The sample function values fp can then be used to estimate alm. A straightforward estimator is

$\displaystyle \hat{a}_{lm}$ $\textstyle \myequal $ $\displaystyle \frac{4\pi}{N_{\rm {pix}}}\sum_{p=0}^{N_{\rm {pix}}-1}
Y^\ast_{lm}(\gamma_p) f(\gamma_p),$ (2)

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.

Eric Hivon 2010-06-18
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