幂级数何时可以解析延拓？
When can a power series be analytically continued?
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中文总结 AI 辅助
本文综述了通过系数解析插值刻画幂级数解析延拓的经典结果，统一符号并重点处理了Carlson第二延拓定理及其基于拉普拉斯变换的改进。
中文摘要 AI 辅助
一个经典的研究方向通过系数解析插值来刻画幂级数的解析延拓。在适当假设下，插值函数的增长反映了延拓域的几何性质，反之亦然。我们综述了从Leau和Le Roy到Lindelöf、Carlson、Dufresnoy–Pisot和Arakelian的这些结果的发展。我们用复平面紧化统一符号表述主要结果。特别地，我们全面处理了Carlson第二延拓定理，该定理此前很少受到关注，并基于拉普拉斯变换证明了其一个改进版本。
英文摘要
A classical line of research characterizes analytic continuation of power series through analytic interpolation of their coefficients. Under suitable hypotheses, the growth of the interpolating function reflects the geometry of the continuation domain, and conversely. We survey the development of these results from Leau and Le Roy through Lindelöf, Carlson, Dufresnoy$\unicode{x2013}$Pisot and Arakelian. We formulate the main results in a unified notation using compactifications of the complex plane. In particular, we give a comprehensive treatment of Carlson's second continuation theorem, which has received little attention, and prove a refinement based on the Laplace transform.
