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

DOI:
https://doi.org/10.1090/S0002-9939-98-04373-1

MathSciNet review:
1452793

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $t>0$.

- N. I. Akhiezer,
*The Classical Moment Problem*, Oliver & Boyd, Edinburgh and London, 1965. - T. Bisgaard,
*Positive definite operator sequences*, Proc. Amer. Math. Soc.**121**(1994), no. 4, 1185–1191. MR**1197531**, DOI https://doi.org/10.1090/S0002-9939-1994-1197531-3 - T. M. Bisgaard and Z. Sasvári, On the positive definiteness of certain functions,
*Math. Nachr.***186**(1997), 81–99. - R. P. Boas, The Stieltjes moment problem for functions of bounded variation,
*Bull. Amer. Math. Soc.***45**(1939), 399–404.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
43A35,
44A60,
47-xx,
60-xx

Retrieve articles in all journals with MSC (1991): 43A35, 44A60, 47-xx, 60-xx

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