Closed dlfivefifty closed 3 months ago
There are "anti-Gaussian" quadrature methods which as far as I know interlace the points of Gaussian quadrature. Not sure if that is why you said "anti-Gauss–Lobatto" or if you just meant "the opposite of what Gauss–Lobatto does".
I meant "opposite of what Gauss-Lobatto" (or Radau): instead of insisting on a specific point, I want to forbid it.
Note its a bit delicate as for even points we get two points tending to (but not exactly) zero
Is there a characterization of the non-negative weights that live in the kernel of the Hilbert transform?
In the even case, does the pair of Gauss points breach |x| = ρ
?
Is there a characterization of the non-negative weights that live in the kernel of the Hilbert transform?
I think the kernel is only |x|/(sqrt(1-x^2)sqrt(ρ^2-x^2))
.
In the even case, does the pair of Gauss points breach |x| = ρ?
Not even close:
julia> golubwelsch(Q[:,Base.OneTo(2)])[1]
2-element Vector{Float64}:
-0.3354101966249685
0.3354101966249685
julia> golubwelsch(Q[:,Base.OneTo(4)])[1]
4-element Vector{Float64}:
-0.8260064989841912
-0.10635236062957643
0.10635236062957698
0.8260064989841945
julia> golubwelsch(Q[:,Base.OneTo(6)])[1]
6-element Vector{Float64}:
-0.9146586174230956
-0.6894624123565193
-0.03501609669916683
0.0350160966991655
0.6894624123565204
0.9146586174230971
That's interesting: I can see why the odd symmetry forces a root at the non-negligible Dirac point mass, but had no idea that it would be "strong" enough to pull other points into zero-weight territory (in the even case).
It sort of makes sense; since one typically expects finite section to converge
Since two bands were moved out of the package I guess it's fine to close this for now.
The two-band analogue of Chebyshev T is now implemented as
T = TwoBandJacobi(ρ, -1/2, -1/2, 1/2)
, orthogonal w.r.t.|x|/(sqrt(1-x^2)sqrt(x^2-ρ^2))
. It has the nice property that the Hilbert transform of its weight is 0, so the Hilbert transform is a simple map fromT
toQ = Associated(T)
, the associated orthogonal polynomials.The surprising thing is
Q
has an orthogonality measure with a delta function at 0, I believe its something likesqrt(1-x^2)sqrt(x^2-ρ^2)/|x| + ρ δ(x)
(where the first term is only valid forρ < |x| <1
). This has the annoying feature that Golub--Welsch gives extra points associated with theδ
:Anyone seen anything like this before and know if there's a modification to Gauss quadrature, i.e. an anti-Gauss–Lobatto that prevents certain points?
I just need this for the expansion in
Q
. I can work around it by using a Vandermonde matrix. But it's an interesting problem.@MikaelSlevinsky @TSGut @ajt60gaibb @marcusdavidwebb