Stieltjes moment sequences and positive definite matrix sequences

Bisgaard, Torben

doi:10.1090/S0002-9939-98-04373-1

Stieltjes moment sequences and positive definite matrix sequences
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by Torben Maack Bisgaard PDF

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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