Search

Start Over Subject positive definite

1. An example of a positive semidefinite double sequence which is not a moment sequence

Creator:
Bisgaard, Torben Maack
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
double sequence, positive definite, and moment sequence
Language:
English
Description:
The first explicit example of a positive semidefinite double sequence which is not a moment sequence was given by Friedrich. We present an example with a simpler definition and more moderate growth as $(m, n) \rightarrow \infty $.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. On the Stieltjes moment problem on semigroups

Creator:
Bisgaard, Torben Maack
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
semigroup, abelian, commutative, finitely generated, positive definite, completely positive definite, and character
Language:
English
Description:
We characterize finitely generated abelian semigroups such that every completely positive definite function (a function all of whose shifts are positive definite) is an integral of nonnegative miltiplicative real-valued functions (called nonnegative characters).
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. Stieltjes perfect semigroups are perfect

Creator:
Bisgaard, Torben Maack and Sakakibara, Nobuhisa
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
perfect, Stieltjes perfect, moment, positive definite, conelike, semi-$*$-divisible, and $*$-semigroup
Language:
English
Description:
An abelian $*$-semigroup $S$ is perfect (resp. Stieltjes perfect) if every positive definite (resp. completely so) function on $S$ admits a unique disintegration as an integral of hermitian multiplicative functions (resp. nonnegative such). We prove that every Stieltjes perfect semigroup is perfect. The converse has been known for semigroups with neutral element, but is here shown to be not true in general. We prove that an abelian $*$-semigroup $S$ is perfect if for each $s \in S$ there exist $t\in S$ and $m,n\in \mathbb{N}_0$ such that $m+n\ge 2$ and $s+s^ *=s^*+mt+nt^*$. This was known only with $s=mt+nt^*$ instead. The equality cannot be replaced by $s+s^*+s=s+s^*+mt+nt^*$ in general, but for semigroups with neutral element it can be replaced by $s+p(s+s^*)=p(s+s^*)+ mt+nt^*$ for arbitrary $p\in \mathbb{N}$ (allowed to depend on $s$).
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public