Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Kuttner’s problem and a Pólya type criterion for characteristic functions
HTML articles powered by AMS MathViewer

by Tilmann Gneiting PDF
Proc. Amer. Math. Soc. 128 (2000), 1721-1728 Request permission


Let $\varphi : [0,\infty ) \to \mathbb {R}$ be a continuous function with $\varphi (0) = 1$ and $\lim _{t \to \infty } \varphi (t)$ $= 0$. If $t^{-1} (\sqrt {t} \varphi ''(\sqrt {t}) - \varphi ’(\sqrt {t}))$ is convex, then $\psi (t) = \varphi (|t|)$, $t \in \mathbb {R}$, is the characteristic function of an absolutely continuous probability distribution. The criterion complements Pólya’s theorem and applies to characteristic functions with various types of behavior at the origin. In particular, it provides upper bounds on Kuttner’s function $k(\lambda )$, $\lambda \in (0,2)$, which gives the minimal value of $\kappa$ such that $(1-|t|^\lambda )_+^\kappa$ is a characteristic function. Specifically, $k(5/3) \leq 3$. Furthermore, improved lower bounds on Kuttner’s function are obtained from an inequality due to Boas and Kac.
Similar Articles
Additional Information
  • Tilmann Gneiting
  • Affiliation: Department of Statistics, Box 354322, University of Washington, Seattle, Washington 98195
  • Email:
  • Received by editor(s): July 13, 1998
  • Published electronically: October 27, 1999
  • Communicated by: Christopher D. Sogge
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 128 (2000), 1721-1728
  • MSC (1991): Primary 42A82, 60E10; Secondary 42A24, 42A38
  • DOI:
  • MathSciNet review: 1646306