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Stieltjes moment sequences
and positive definite matrix sequences

Author: Torben Maack Bisgaard
Journal: Proc. Amer. Math. Soc. 126 (1998), 3227-3237
MSC (1991): Primary 43A35, 44A60, 47-xx, 60-xx
MathSciNet review: 1452793
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Abstract: For a certain constant $\delta>0$ (a little less than $1/4$), every function $f\colon {\mathbb N}_0\to \left]0,\infty\right[$ 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 $\varphi\colon {\mathbb N}_0\to \left]0,\infty\right[$ 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 $t>0$.

References [Enhancements On Off] (What's this?)

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  • 3. T. M. Bisgaard and Z. Sasvári, On the positive definiteness of certain functions, Math. Nachr. 186 (1997), 81-99. CMP 97:16
  • 4. R. P. Boas, The Stieltjes moment problem for functions of bounded variation, Bull. Amer. Math. Soc. 45 (1939), 399-404.

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Additional Information

Torben Maack Bisgaard
Affiliation: Nandrupsvej 7 st. th., DK-2000 Frederiksberg C, Denmark

Keywords: Stieltjes moment sequence, indeterminate, moment sequence, positive definite, positive type, convolution semigroup
Received by editor(s): July 15, 1996
Received by editor(s) in revised form: February 24, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society