A new class of bell-shaped functions
We provide a large class of functions f that are bell-shaped: the nth 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 funct...
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| Veröffentlicht in: | Transactions of the American Mathematical Society 2020-04, Vol.373 (4), p.2255-2280 |
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| container_title | Transactions of the American Mathematical Society |
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| creator | Mateusz Kwaśnicki |
| description | We provide a large class of functions f that are bell-shaped: the nth 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. |
| doi_str_mv | 10.1090/tran/7825 |
| format | Article |
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| title | A new class of bell-shaped functions |
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