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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A new class of bell-shaped functions
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by Mateusz Kwaśnicki PDF
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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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
  • 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
  • © Copyright 2020 American Mathematical Society
  • 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
  • MathSciNet review: 4069218