格林函数的退化行为
Degenerating behavior of Green's function
摘要：
设实区间$I = \cup_{j = 1}^l [a_{2 j -1},a_{2j}],$ $a_1 < ... < a_{2 l},$和$I_n = \cup_{k = 1}^m [B_{k,n}, C_{k,n}]$的并集是这样的，即$\cap_{k = 1}^{\infty} [B_{k,n},C_{k,n}] = \{c_k \}$对于$k = 1,...,m$和${\rm dist}(E,I_n) \geq const > 0.$我们展示了如何渐近地表示在$z = \infty$处的$E \cup I_n$的格林函数$ϕ(z,\infty,E \cup I_n)$在格林函数$ϕ(z,\infty,E)$和$ϕ(z,c_k,E).$方面该公式产生$ϕ^n(z,\infty,E \cup I_n)$关于$n$的立即渐近性，这在逼近论的许多问题中是重要的。另一个结果是用$cap(E)$和$ϕ(z,c_k,E)$以及调和测度$ω(\infty, E_j,E \cup I_n).$表示的$cap(E \cup I_n)$的渐近表示
英文摘要：
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 $ω(\infty, E_j,E \cup I_n).$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
