Degenerating behavior of Green's function

Details Degenerating behavior of Green's function Degenerating behavior of Green's function

2010-01-04

arxiv.org/abs/1001.0485v1

Mathematics

Complex Variables

The manuscript was prepared by the author in the two months preceding his passing away in November 2009. The manuscript remain

Scientific paper

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

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