论文标题
某些紧凑的平面子集的平均有理近似
Mean Rational Approximation for Some Compact Planar Subsets
论文作者
论文摘要
1991年，J。Thomson获得了$ p^t（μ）的著名结构结果。结果表明，如果$ r^t（k，μ）$是纯净的，则$ r^t（k，μ）\ cap l^\ infty（μ）$是“与$ \ mbox {abpe}（abpe}（r^t（k，k，k，μ）），$ box的分析函数的代数”，$ r^t（k，μ）$。我们表明，如果允许$ \ mathbb c \ setMinus k $的成分直径倾向于零，那么即使$ \ text {int}（k）= \ mbox {abpe}（abpe}（r^t（k，k，μ））$ and $ k = \ k = \ ypeck = \ text {\ text {\ text {int} $^k， \ cap l^\ infty（μ）$可能“等于” $ \ text {int}（k）上有界分析函数的适当的子代数，$k。$ $K。$ $K。$的某些部分的函数在$k。$的某些部分上是“连续”的$。
In 1991, J. Thomson obtained celebrated structural results for $P^t(μ).$ Later, J. Brennan (2008) generalized Thomson's theorem to $R^t(K,μ)$ when the diameters of the components of $\mathbb C\setminus K$ are bounded below. The results indicate that if $R^t(K,μ)$ is pure, then $R^t(K,μ) \cap L^\infty (μ)$ is the "same as" the algebra of bounded analytic functions on $\mbox{abpe}(R^t(K, μ)),$ the set of analytic bounded point evaluations. We show that if the diameters of the components of $\mathbb C\setminus K$ are allowed to tend to zero, then even though $\text{int}(K) = \mbox{abpe}(R^t(K, μ))$ and $K =\overline {\text{int}(K)},$ the algebra $R^t(K,μ) \cap L^\infty (μ)$ may "be equal to" a proper sub-algebra of bounded analytic functions on $\text{int}(K),$ where functions in the sub-algebra are "continuous" on certain portions of the inner boundary of $K.$
