平坦函数之高阶导数的估计及应用

Estimation of Higher-Order Derivatives of Flat Functions and its Application

研究平坦函数n阶导函数的上确界范数.基于泛函分析的思想,使用基本的数学分析知识,证明了上确界范数之渐近增长快于n的阶乘,并用此结论重新证明了完全单调函数的解析性.文末还阐释了该渐近估计在一定意义下达到了最优.

In this paper,we investigate the sup-norm of n-th derivative for any flat function.Based on ideas from functional analysis,using some basic knowledge of mathematical analysis,we prove that the asymptotic growth of sup-norms is faster than n!.As an application of this result,we give a new proof of the analyticity of completely monotonic functions.In the end of this paper we also show that our asymptotic estimation is optimal in some sense.