Definition of stieltjes? #39

Activity

1. dlfivefifty commented on Apr 20, 2024

   

   dlfivefifty

   on Apr 20, 2024

   Last edited by dlfivefifty

   Member

   More actions

   When you use the full broadcast notation it's clear what the domain is:

   W = Weighted(ChebyshevT())
   x = axes(W,1)
   S = inv.(x .- x') * W
   @test axes(S) == (Inclusion(-1..1), 1:∞)

   So I guess you are referring to the behaviour of stieltjes(W)? I think the convention is that it evaluates on the domain axes(W,1).

   This may also be true for complex supported domains but I've just realised if I write

   Γ = somecomplexcontour
   W = Weighted(chebyshevt(Γ))
   z = axes(W,1)
   S = inv.(z .- z') * W

   The definition follows from the definition of the adjoint/inner product. Which would only sensibly be the $\int_Γ \bar f(z) g(z) ds$ where we would also want a version with $dz$. Probably we would use stieltjes to mean the latter and cauchy (with the $1/(2πi)$ to mean the former.

   If you indeed want the whole real line we can support the syntax

   W = Weighted(ChebyshevT())
   x = Inclusion(ℝ)
   t = axes(W,1)
   S = inv.(x .- t') * W
   @test axes(S) == (x, 1:∞)

   which could lower to a call steiltjes(W, ℝ). Would take some thought how to implement this.