Degenerating behavior of Green's function

Abstract

Let the unions of real intervals $I={\cup }_{j=1}^l[a_{2j-1},a_{2j}],$ $a_1<...<a_{2l},$ 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 $\mathrm{dist}(E,I_n)\ge const>0.$ We show how to express asymptotically the Green's function $\varphi (z,\infty ,E\cup I_n)$ of $E\cup I_n$ at $z=\infty$ in terms of the Green's function $\varphi (z,\infty ,E)$ and $\varphi (z,c_k,E).$ The formula yields immediately asymptotics for ${\varphi }^n(z,\infty ,E\cup I_n)$ 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 $\varphi (z,c_k,E)$ and of the harmonic measure $\omega (\infty ,E_j,E\cup I_n).$

The manuscript was prepared by the author in the two months preceding his passing away in November 2009. The manuscript remained unsubmitted and is not published elsewhere