Degenerating behavior of Green's function

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 $\omega(\infty, E_j,E \cup I_n).$