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### HEALpix and spherical harmonic sampling

Posted: October 24 2004
Hi,

I have still a question about HEALpix grid in the particular context of spherical harmonic (SH) sampling and SH transform.

Are we sure that SH transform can be exactly realized to all orders exactly as it can be done on traditional equi-angular grid (I mean, D theta = cste, D phi = cste)?

The problem is that, in polar caps, the number of points on each iso-latitude ring decreases as its distance to the North pole, i.e. from 4*Nside to 4. However, spherical harmonics Y_l^m of order (l,m) (with m<=|l|) contain the term exp(i m phi) which needs at least m+1 points to be correctly sampled (without aliasing) on each iso-latitude ring.

So, I'm not sure that Spherical Fourier transforms, i.e. spherical harmonic transform, can be realized exactly on HEALpix grid. In best cases, this can be just an approximation with a certain error.

What do you think ? Perhaps it is already well known but I haven't read such remark elsewhere. Do you know a reference which explains that effect ?

Best,
Laurent.

### Healpix sampling

Posted: October 27 2004
Yes, it is approximate.

However near the poles the spherical harmonics go like

$$Y_{lm} \sim (-1)^m \frac{e^{im\phi}}{m!}\sqrt{\frac{(2l+1)}{4\pi} \frac{(l+m)!}{(l-m)!}} \left( \frac{\theta}{2}\right)^m$$

so for large m and small $$\theta$$ the contributions go like $$\theta^m$$ which is tiny. So although HealPix does include all $$m\le l$$, near the polar caps you can to very good accuracy neglect high-m contributions (much better than numerical precision).

This makes sense because $$e^{i\ell\phi}$$ has a spatial frequency much higher near the poles than at the equator, so you wouldn't expect a significant contribution near the poles unless m is small compared to $$\ell$$. So a small number of pixels near the poles is not so bad.

Posted: October 28 2004
However, IMHO, approximation is not so harmless since for instance for $$i = N_{\rm side}/2$$ (i.e. in the North cap and with $$i$$ the iso-latitude ring label), spherical harmonics are so not well estimated by $$C e^{{\rm i} m\phi} \big(\frac{\theta}{2}\big)^m$$, leading to more important contributions and preventing good computations of SH transform on $$|m|\geq N_{\rm side}/2$$ in case where $$l\geq N_{\rm side}/2$$ (for high frequencies).
From the differential equation for the $$P_{lm}$$ you can show that the turning point (where the character changes from oscillatory to decaying) is at $$\sin\theta \sim m/\ell$$. Hence for $$m >> \ell \sin\theta$$ you are always well in the damped tail, and the contributions are small.