Degenerating behavior of Green's function

الأصل · EN

Let the unions of real intervals I = ∪ⱼ ₌ ₁ˡ [a₂ ⱼ ₋₁,a₂ⱼ], a₁ <... < a₂ ₗ, and Iₙ = ∪ₖ ₌ ₁ᵐ [Bₖ,ₙ, Cₖ,ₙ] be such that ∩ₖ ₌ ₁∞ [Bₖ,ₙ,Cₖ,ₙ] = {cₖ } for k = 1,...,m and dist(E,Iₙ) ≥ const > 0. We show how to express asymptotically the Green's function ϕ(z,∞,E ∪ Iₙ) of E ∪ Iₙ at z = ∞ in terms of the Green's function ϕ(z,∞,E) and ϕ(z,cₖ,E). The formula yields immediately asymptotics for ϕⁿ(z,∞,E ∪ Iₙ) with respect to n which are important in many problems of approximation theory. Another consequence is an asymptotic representation of cap(E ∪ Iₙ) in terms of cap(E) and ϕ(z,cₖ,E) and of the harmonic measure ω(∞, Eⱼ,E ∪ Iₙ).