Improve FFT interpolation (1/2)

[unalmis]

This is the first of two pull requests to improve the FFT interpolation.

4 goals are achieved by this PR.

• The real FFT is now used where possible. This is especially useful for 2D transforms. Resolves Use real fft in interpolation #75 .
• Double the width of the Fourier spectrum is now preserved when interpolating to a less dense grid, at no additional cost. Previously, when the number of requested output samples $n_{\text{out}}$ was less than the number of input samples $n_{\text{in}}$, the interpolation would neglect the $n_{\text{in}} - n_{\text{out}}$ highest frequency spectral coefficients, resulting in lossy interpolation. Now the interpolation is lossless for all $n_{\text{out}} >= \lfloor n_{\text{in}} / 2 \rfloor + 1$. Likewise, if $n_{\text{out}} < \lfloor n_{\text{in}} / 2 \rfloor + 1$, then only the $\lfloor n_{\text{in}} / 2 \rfloor + 1 - n_{\text{out}}$ highest frequency spectral coefficients will be neglected.
• In the 2D upsampling case, the second transform is now padded only after computing the first transform. This reduces the size of the problem, so the computation is less expensive.
• In the 2D downsampling case, the second transform is now truncated prior to computing the first transform. This reduces the size of the problem, so the computation is less expensive.

[f0uriest]

doesn't this truncation mean that we're still interpolating the band limited signal, not the full one?

[unalmis]

c.shape[0] has size f.shape[0]//2 + 1. so our band limited signal can preserve double the width of the fourier spectrum than before

[f0uriest]

can you explain this part? Isn't this just multiplying the output by exp(i*n*x)?

[unalmis]

we transform sum_n^N c_n e^(i n x) for n>=0 to e^iNx//2 sum_k c_n e^(ikx) where k now includes the relevant negative values (that result from pulling the highest frequency outside the sum)

[unalmis]

https://github.com/PlasmaControl/DESC/blob/82d2fe6221976e93c54415195e32e62bfe07d1a7/tests/test_integrals.py#L584

[f0uriest]

What's the status of this? It looks like there are some merge conflicts. Previously the method was only correct for real valued data, but I've since fixed it so that it works for complex data as well, but I'd be happy to include this as a special case when the data is purely real

[unalmis]

but I've since fixed it so that it works for complex data as well, but I'd be happy to include this as a special case when the data is purely real

The improvements in points 3 and 4 in the PR header would still apply to complex transforms.

It looks like there are some merge conflicts.

The merge conflicts are resolved by updating the names of the functions and tests here now that the original ones support complex data. interpax-fft here simply includes #117 with labels rfft instead of fft. As I suspect you may have preferences on API/names, e.g. whether to detect if data is real and call the private funs, I think you should just update and push changes here to your preference.

[unalmis]

yea so just add interpax-fft as a depedency and call it if input data is not jnp.complexobj