关于光滑函数的Taylor系数
On Taylor coefficients of smooth functions
摘要：
研究了Borel映射，它将区间上的无穷可微函数映射到区间上某点的Taylor系数的喷注上。我们的主要结果包括一个完整的描述的图像的博雷尔映射的Beurling类的光滑函数和一个时刻型求和方法，它允许一个恢复功能从泰勒喷气。这种描述的一个令人惊讶的特征是对数类的意外阈值。另一个有趣的发现是一个“对偶”之间的非准解析和准解析类，这减少了描述的图像的Borel映射的非准解析类的一个相应的准解析类，并补充经典的结果Carleson和Escherapreis。
英文摘要：
We study the Borel map, which maps infinitely differentiable functions on an interval to the jets of their Taylor coefficients at a given point in the interval. Our main results include a complete description of the image of the Borel map for Beurling classes of smooth functions and a moment-type summation method which allows one to recover a function from its Taylor jet. A surprising feature of this description is an unexpected threshold at the logarithmic class. Another interesting finding is a "duality" between non-quasianalytic and quasianalytic classes, which reduces the description of the image of the Borel map for non-quasianalytic classes to the one for the corresponding quasianalytic classes, and complements classical results of Carleson and Ehrenpreis.
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