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Let's implement this stepwise, starting at the most simple (inefficient) implementation and then building up the to the more complicated (efficient) implementation. Steps should include the following:

• Direct
• + Separation of variables
• + FFT

Note the the full MW inverse transform with be considered in a separate issue (#17).

We should support Wigner transforms, i.e. Fourier transforms on SO(3), soon so that they can be used by s2wav. In the first instance these can be pure Python and don't need to be JAX'ified.

There are various ways to do this but the simplest is to implement what is described in this paper (See Section V). Then we can just use the spin transforms already implemented and simply need to add an extra FFT over gamma.

Avoid using FFT in rings as a validation

At present not clear if modules are appearing in the documentation.

Also, let's deploy docs build from main to github branches. That will be public but that's not a problem.

• Set up wrapper functions for Trapani implementations
• Document JAX implementations in Trapani module

JAX implementations no not necessarily run on the GPU. Our code should be device agnostic. So we should distinguish JAX or not. If a GPU is available JAX can run on that, otherwise it will run on a CPU. (Also, JAX can use TPUs, so we could potentially run on TPUs.)

Let's implement this stepwise, starting at the most simple (inefficient) implementation and then building up the to the more complicated (efficient) implementation. Steps should include the following:

• Direct
• + Separation of variables
• + FFT

At present we have a flip in the Wigner tests here when comparing the Turok slice approach to ssht. Why is that?

This may also relate to this comment in the docstring. We need to check the comment is correct (perhaps the ordering could be different).

I think we should adopt the MIT licence. We want this code to be very widely used so a permissible licence is probably best (rather than, e.g., GPL).

The operator method computes all m, m', for a given el and theta. There is no recurrsion over el. So the method is very well suited to acceleration.

See implementing in ssht here.

Currently the classifiers in setup.py (weakly) suggest Python 3.6, 3.7 and 3.8 are supported

while the tests are only run under Python 3.8 in the GitHub Actions continuous integration

Currently the tests fail when trying to run in a Python 3.6 environment (I think due to use of Python 3.7+ features in JAX) but do pass under Python 3.7.

I would suggest following NEP 29 in deciding which Python and NumPy versions to target, which would currently suggest supporting Python 3.8+ and NumPy 1.20+. It would be good to settle on a minimum version to target so we know what language features its safe to use.

The main concrete thing this would require changing is to add a python_requires argument to setup in the setup.py script and making sure the Python version(s) we are testing against matches this.

We should double check we are consistent everywhere since in the 2d flm revisions we updated some of the healpix indexing to note that lmax = L - 1.

Implement full MW inverse transform for complex signals. See naive implementations of #16 for comparisons.

Functions to compute vectors theta/phi in RING ordering.

Basically just install s2fft. No development really required for this but I thought I'd nevertheless capture this in an issue. Just shout if you have any problems or questions!

This would avoid the current L-1 offsets.