LAUSR

A nonnegative real function f is bell-shaped if it converges to zero at ±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm \infty $$\end{document} and the nth derivative of f changes sign n times for every n=0,1,2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 0, 1, 2, \ldots $$\end{document} Similarly, a nonnegative sequence (a(k):k∈Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a(k): k \in \mathbb {Z})$$\end{document} is bell-shaped if it converges to zero at ±∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm \infty $$\end{document} and the nth iterated difference of a(k) changes sign n times for every n=0,1,2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n = 0, 1, 2, \ldots $$\end{document} A characterisation of bell-shaped functions was given by Thomas Simon and the first named author, and recently a similar result for one-sided bell-shaped sequences was found by the authors. In the present article we give a complete description of two-sided bell-shaped sequences. Our main result proves that bell-shaped sequences are convolutions of Pólya frequency sequences and what we call absolutely monotone-then-completely monotone sequences, and it provides an equivalent, and relatively easy to verify, condition in terms of holomorphic extensions of the generating function. We also prove that if f is a bell-shaped function, then f(k) is a bell-shaped sequence.