### 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

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.

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.