关于Borel可和性与解析泛函
On Borel summability and analytic functionals
摘要：
证明了形式幂级数有正的收敛半径当且仅当它在圆心为原点的圆上是一致Borel可和的。因此，我们得到了整函数$f$是指数型的当且仅当形式幂级数$\sum_{n=0}^{\infty}f^{(n)}(0)z^{n}$在以原点为中心的圆上是一致Borel可和的.我们应用这些结果，以获得这些席尔瓦回火超分布，这是解析泛函的表征。我们还使用Borel求和表示解析泛函的Borel和他们的时刻泰勒级数在Borel多边形。
英文摘要：
We show that a formal power series has positive radius of convergence if and only if it is uniformly Borel summable over a circle with center at the origin. Consequently, we obtain that an entire function $f$ is of exponential type if and only if the formal power series $\sum_{n=0}^{\infty}f^{(n)}(0)z^{n}$ is uniformly Borel summable over a circle centered a the origin. We apply these results to obtain a characterization of those Silva tempered ultradistributions which are analytic functionals. We also use Borel summability to represent analytic functionals as Borel sums of their moment Taylor series over the Borel polygon.
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