Quadratically shifted bases

[TSGut]

I want to use $P(2x^2-1)$. I had this working when the repo was still OrthogonalPolynomialsQuasi.jl but some changes since have stopped it from working with my setup and I don't want to use the old repo. I think this should just be easy to do out of the box in this package, especially given how ubiquitous $P(2x^2-1)$ is for rotationally symmetric bases.

It seems a Chebyhev quadratic shift is implemented in the tests, see

ClassicalOrthogonalPolynomials.jl/test/test_chebyshev.jl

Line 478 in 139fee6

struct QuadraticMap{T} <: Map{T} end

I think this should be low effort to get working for at least general Jacobi polynomials. Any objections to me adding this functionality to the package itself rather than just have it defined in tests?

Once we do this I would also be happy to type up some documentation on shifted bases. 😁

[dlfivefifty]

Are mappings really the right way to do this? It might be better to think of these as SymmetricLegendre living in SymmetricOrthogonalPolynomials.jl as then the symmetry aspect is explicit....

[TSGut]

Do you mean "the right way to do this" in terms of package placement or in terms of the underlying implementation? Implementation wise I think it's a pretty solid way of doing it. I'm open to better ideas if that is what you meant.

Basically I'm trying to update and open source our higher dimensional equilibrium measures stuff in something like EquilibriumMeasures.jl but I need these bases working before I can do that. Making this depend on a for now private repo would complicate things, although I do agree that it would make sense for these radially symmetric bases to live in SymmetricOrthogonalPolynomials.jl eventually.

So if that is the plan, then for now I can just have the required bases explicitly coded into EquilibriumMeasures.jl until SymmetricOrthogonalPolynomials.jl is more mature and then move it there?

[dlfivefifty]

Yes I think that's sensible... I think it would be a mistake to widen the scope of this package TBH

[TSGut]

Great, I'll start work on including it in EquilibriumMeasures.jl for now and will open an issue in SymmetricOrthogonalPolynomials.jl to remind myself to add it there when we know more about how that package will be organized.