Intrinsic interpolation, near-circularity and maximal convergence - Blatt, Hans-Peter (Journal of approximation theory, 2025)
Let E be compact and connected with capE>0 and connected complement Ω=ℂ¯∖E, let gΩ(z,∞) be the Green’s function of Ω with pole at infinity and let Eσ≔{z∈Ω:gΩ(z,∞)<logσ}∪E,1<σ<∞, be the Green domains with boundaries Γσ. Let f be holomorphic on E and let ρ(f) denote the maximal parameter of holomorphy of f and let pnn∈N be a sequence of polynomials converging maximally to f on E. If σ, 1<σ<ρ(f)<∞, is fixed and if mn(σ) denotes the number of interpolation points of pn to f in Eσ with normalized counting measure μσ,n, then there exists a subset Λ⊂N such that mn(σ)=n+o(n)asn∈Λ,n→∞,μσ,n|Ê+μσ,n|Ω⟶∗μEasn∈Λ,n→∞, where μσ,n=μσ,n|E+μσ,n|Ω, μσ,n|Ê denotes the balayage measure of μσ,n|E onto the boundary of E and μE is the equilibrium measure of E. Moreover, there exists a sequence σnn∈Λ converging to σ such that the closed curves γn=(f−pn)(Γσn) do not pass through the point 0 and the winding numbers Indγn(0) satisfy Indγn(0)=mn(σn)=n+o(n)asn∈Λ,n→∞.