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arXiv · 1001.0485

Degenerating behavior of Green's function

Franz Peherstorfer
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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 $ϕ(z,\infty,E \cup I_n)$ of $E \cup I_n$ at $z = \infty$ in terms of the Green's function $ϕ(z,\infty,E)$ and $ϕ(z,c_k,E).$ The formula yields immediately asymptotics for $ϕ^n(z,\infty,E \cup I_n)$ 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 $ϕ(z,c_k,E)$ and of the harmonic measure $ω(\infty, E_j,E \cup I_n).$

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Franz Peherstorfer. 2010-01-04. Degenerating behavior of Green's function. https://arxiv.org/abs/1001.0485

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