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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A new class of bell-shaped functions


Author: Mateusz Kwaśnicki
Journal: Trans. Amer. Math. Soc. 373 (2020), 2255-2280
MSC (2010): Primary 26A51, 60E07; Secondary 60E10, 60G51
DOI: https://doi.org/10.1090/tran/7825
Published electronically: January 28, 2020
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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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Additional Information

Mateusz Kwaśnicki
Affiliation: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: mateusz.kwasnicki@pwr.edu.pl

DOI: https://doi.org/10.1090/tran/7825
Keywords: Bell-shape, P\'olya frequency function, completely monotone function, absolutely monotone function, Stieltjes function, generalised gamma convolution
Received by editor(s): February 23, 2018
Received by editor(s) in revised form: November 26, 2018, and January 9, 2019
Published electronically: January 28, 2020
Additional Notes: This work was supported by the Polish National Science Centre (NCN) grant no.2015/19/B/ST1/01457
Dedicated: In memory of Augustyn Kałuża
Article copyright: © Copyright 2020 American Mathematical Society