Characterisation of the class of bell-shaped functions

A non-negative function $f$ is said to be ‘bell-shaped’ if $f$ tends to zero at $\pm \mathrm{\infty }$ and the $n$-th derivative of $f$ changes its sign $n$ times for every $n=0,1,2,\dots$ 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 $n\to \mathrm{\infty }$.