A new class of bell-shaped functions

Kwaśnicki, Mateusz

doi:10.1090/tran/7825

A new class of bell-shaped functions
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Trans. Amer. Math. Soc. 373 (2020), 2255-2280 Request permission

Abstract:

We provide a large class of functions $f$ that are bell-shaped: the $n$th derivative of $f$ changes its sign exactly $n$ times. This class is described by means of Stieltjes-type representation of the logarithm of the Fourier transform of $f$, and it contains all previously known examples of bell-shaped functions, as well as all extended generalised gamma convolutions, including all density functions of stable distributions. The proof involves representation of $f$ as the convolution of a Pólya frequency function and a function which is absolutely monotone on $(-\infty , 0)$ and completely monotone on $(0, \infty )$. In the final part we disprove three plausible generalisations of our result.

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