This might actually be Differential Geometry but I'll post it here for now.

I was looking at a paper yesterday about quantization conditions when working on a manifold based on the surface of a sphere. I've never seen that before and it's apparently a lot less intuitive than I would have thought. The paper mentions a "generalized Fourier transform." The best I could do on an internet search is

here. But I don't think that's what the paper had in mind because the Wolfram link looks like it's modeled in the usual 3-space, not a more generalized manifold.

Has anyone heard of this? I can quote some sections of the paper if you need it.

Thanks!

-Dan

OK having read the paper, here is my understanding of the situation.

The classic Fourier Transform is based on functions F : R → R and integrals which converge such as the Fourier sine or cosine transforms.

The first level of generalisation was to complex domains which also allowed divergent (infinite) intergals to be taken on board.

This is the level that is mostly used by Physicists and Engineers.

The level allowed the connection to Green's functions and the solution of the simplest forms of Helmholtz equation, amongst other things.

The introduction of Quantum Theory, and in particular the Dirac Delta function necessitated a further generalisation and came at the time of the theory of distributions or generalised functions which supplanted the older real and complex analysis.

The second level came with generalised Green's functions and Helmholtz equations and in theory is more or less the level of generalisation in the paper.

A third level of generalisation came linking Topology and algebra and this is where the Group theory and Noether references arise.

I have about twenty books incorporating the development of these levels and their applications in such wide fields as Functional Analysis, Operational methods, Partial Differential eaqutions, Applied Maths.

Here is a short introduction to the Fourier Transforms of Distributions from one of them on Functional analysis.