# 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.

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

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

## Abstract:

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.
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