钟形函数类的刻画
Characterisation of the class of bell-shaped functions
摘要：
一个非负函数$f$被称为“钟形”，如果$f$趋于零，$f$和$f$的$n$次导数改变其符号$n$倍，每$n = 0, 1, 2, \ldots$我们提供了一个完整的特征的类钟形函数：我们证明，每一个钟形函数是卷积的“波利亚频率函数”和一个 * 绝对单调，然后完全单调 * 功能。给出了傅里叶变换全纯延拓的一个等价条件。作为推论，钟形函数的各种性质如下。特别地，我们证明了钟形概率分布是无限可分的，并且钟形函数的$n$阶导数的零点以线性速率增长为$n \to \infty$。
英文摘要：
A non-negative function $f$ is said to be 'bell-shaped' if $f$ tends to zero at $\pm \infty$ and the $n$-th derivative of $f$ changes its sign $n$ times for every $n = 0, 1, 2, \ldots$ We provide a complete characterisation of the class of bell-shaped functions: we prove that every bell-shaped function is a convolution of a 'Pólya frequency function' and an *absolutely monotone-then-completely monotone* function. An equivalent condition in terms of the holomorphic extension of the Fourier transform is also given. As a corollary, various properties of bell-shaped functions follow. In particular, we prove that bell-shaped probability distributions are infinitely divisible, and that the zeroes of the $n$-th derivative of a bell-shaped function grow at a linear rate as $n \to \infty$.
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