2017/10/30 by Kwaśnicki, Mateusz
#Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1710.11023
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 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 (-∞, 0) and completely monotone on (0, ∞). In the final part we disprove three plausible generalisations of our result.