Degenerating behavior of Green's function

TL;DR

This paper derives asymptotic formulas for Green's functions and capacities of unions of real intervals and shrinking sets, with implications for approximation theory.

Contribution

It provides explicit asymptotic expressions for Green's functions and capacities of complex interval unions, linking them to simpler known functions.

Findings

Asymptotic expression for Green's function of union of intervals and shrinking sets.

Asymptotic formulas for capacity and harmonic measure of these unions.

Results applicable to approximation theory problems.

Abstract

Let the unions of real intervals $I = \cup_{j = 1}^l [a_{2 j -1},a_{2j}],$ $a_1 < ... < a_{2 l},$ and $I_n = \cup_{k = 1}^m [B_{k,n}, C_{k,n}]$ be such that $\cap_{k = 1}^{\infty} [B_{k,n},C_{k,n}] = \{c_k \}$ for $k = 1,...,m$ and ${\rm dist}(E,I_n) \geq const > 0.$ We show how to express asymptotically the Green's function $\phi(z,\infty,E \cup I_n)$ of $E \cup I_n$ at $z = \infty$ in terms of the Green's function $\phi(z,\infty,E)$ and $\phi(z,c_k,E).$ The formula yields immediately asymptotics for $\phi^n(z,\infty,E \cup I_n)$ with respect to $n$ which are important in many problems of approximation theory. Another consequence is an asymptotic representation of $cap(E \cup I_n)$ in terms of $cap(E)$ and $\phi(z,c_k,E)$ and of the harmonic measure $\omega(\infty, E_j,E \cup I_n).$