Characterisation of the class of bell-shaped functions

TL;DR

This paper fully characterizes bell-shaped functions, showing they are convolutions of Pólya frequency and monotone functions, and explores their properties including infinite divisibility and zero distribution of derivatives.

Contribution

It provides a complete characterization of bell-shaped functions via convolution with Pólya frequency and monotone functions, and links their properties to Fourier transform extensions.

Findings

Bell-shaped functions are convolutions of Pólya frequency and monotone functions.

Bell-shaped probability distributions are infinitely divisible.

The zeros of derivatives grow linearly with the derivative order.

Abstract

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 'P\'olya 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 $n \to \infty$.