# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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## Stieltjes moment sequences and positive definite matrix sequencesHTML articles powered by AMS MathViewer

by Torben Maack Bisgaard
Proc. Amer. Math. Soc. 126 (1998), 3227-3237 Request permission

## Abstract:

For a certain constant $\delta >0$ (a little less than $1/4$), every function $f\colon \mathbb {N}_0\to ]0, \infty [$ satisfying $f(n)^2\leq \delta f(n-1)f(n+1)$, $n\in \mathbb {N}$, is a Stieltjes indeterminate Stieltjes moment sequence. For every indeterminate moment sequence $f\colon \mathbb {N}_0\to \mathbb {R}$ there is a positive definite matrix sequence $(a_n)$ which is not of positive type and which satisfies $\operatorname {tr}(a_{n+2})=f(n)$, $n\in \mathbb {N}_0$. For a certain constant $\varepsilon >0$ (a little greater than $1/6$), for every function $\varphi \colon \mathbb {N}_0\to ]0, \infty [$ satisfying $\varphi (n)^2\leq \varepsilon \varphi (n-1)\varphi (n+1)$, $n\in \mathbb {N}$, there is a convolution semigroup $(\mu _t)_{t\geq 0}$ of measures on $\mathbb {R}_+$, with moments of all orders, such that $\varphi (n)=\int x^n d\mu _1(x)$, $n\in \mathbb {N}_0$, and for every such convolution semigroup $(\mu _t)$ the measure $\mu _t$ is Stieltjes indeterminate for all $t>0$.
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