Mikael Slevinsky

I was thinking that block banded matrices would lead to fast spectral solutions of Schrödinger eigenproblems on the sphere

One storage scheme for skyline matrices is by a field containing contiguous memory and another containing integers that denote the indices of the beginning of each row/column of data. http://www.netlib.org/utk/people/JackDongarra/etemplates/node378.html....

Hey @mjyshin, you can also do this with a symmetric-definite and banded eigendecomposition when propagating with self-adjoint linear differential operators. A reference on the code below is [here](https://doi.org/10.1016/j.jcp.2020.109383), and this...

1. Yes, Legendre series are required for this method. Not really a big deal since Chebyshev and Legendre series can be converted to and from quite rapidly. 2. I had...

> If we define L using differential operators on, say Chebfun(...), is there a way to convert it to a mapping from Ultraspherical(0.5,...) to the appropriate range? This can be...

It also creates a `RaggedMatrix` instead of a `Diagonal` or `BandedMatrix`. And some Fourier-based spaces (e.g. `SinSpace()`) don't need wrappers for `Derivative`, so the Laplacian shouldn't be a wrapper either.

Yes, the notion of a diagonal operator would be good: some integral operators are diagonal but not a finite sum of Kronecker products. Cheers, Mikael On Oct 16, 2018, at...

Is your issue with this line? https://github.com/JuliaApproximation/ApproxFun.jl/blob/master/src/Spaces/Fourier/Fourier.jl#L172

How do you intend to remove the call to `cos` if it's a cosine series? unless you plan on using Reinsch's modification.

Unless you're referring to mapped domains such as `CosSpace(Circle())`, where the cosine call can be avoided, (essentially, the current implementation to mapping `tocanonical` followed by `cos` could be simplified).