Extreme positive definite double sequences which are not moment sequences
type: Item
instance of: scholarly article
title: Extreme positive definite double sequences which are not moment sequences
zbMATH Open document ID: 1029.43002
author: Torben Maack Bisgaard
published in: Collectanea Mathematica
publication date: 2003-06-30T00:00:00Z
full work available at URL: https://eudml.org/doc/43026
review text: Let \(\left(S,+\right)\) be a countable abelian semigroup with zero. A function \(\varphi: S\to{\mathbb R}\) is positive definite if \[ \sum_{j,k=1}^n c_jc_k\varphi\left(s_j+s_k\right)\geq 0 \] for any \(n\in{\mathbb N}\), \(s_1,\dots, s_n\in S\), and \(c_1,\dots,c_n\in{\mathbb R}\). The set of all positive definite functions is denoted by \(\mathcal{P}\left(S\right)\). A character on \(S\) is a function \(\sigma: S\to{\mathbb R}\) such that \(\sigma\left(0\right)=1\) and \(\sigma\left(s+t\right)=\sigma\left(s\right)\sigma\left(t\right)\) for all \(s,t\in S\). \(S^*\) denotes the set of all characters on \(S\) and it is equipped with the trace topology of the topology of pointwise convergence on \({\mathbb R}^S\). A function \(\varphi: S\to{\mathbb R}\) is called a moment function if there is a Radon measure \(\mu\) on \(S^*\) such that \[ \varphi\left(s\right)=\int_{S^*}\sigma\left(s\right) d\mu\left(\sigma\right), \quad s\in S. \] \(\mathcal{H}\left(S\right)\) denotes the set of all moment functions on \(S\). It is obvious that \(\mathcal{H}\left(S\right)\subset\mathcal{P}\left(S\right)\). If \(\mathcal{H}\left(S\right)=\mathcal{P}\left(S\right)\), the semigroup \(S\) is called semiperfect. As an example of such a group we can use \({\mathbb N}_0=\left\{0,1,2,\dots\right\}\). However, the group \({\mathbb N}_0^k\) is not semiperfect for any \(k\geq 2\). Denote by \(\mathcal{P}_e\left(S\right)\) the set of all generators of extreme rays in \(\mathcal{P}\left(S\right)\). The author remarks that if \(S\) is not semiperfect then the set \(\mathcal{P}_e\left(S\right)\setminus\mathcal{H}\left(S\right)\) must be nonempty. A sequence \(\left\{s_n\right\}_{n=0}^\infty\) of reals is positive definite if and only if it is a moment sequence, i.e., \[ s_n=\int_{\mathbb R}x^n d\mu,\quad n\in{\mathbb N}_0, \] for some measure \(\mu\) on \({\mathbb R}\). The moment sequence \(\left\{s_n\right\}_{n=0}^\infty\) is determinate if there is only one such \(\mu\); otherwise it is indet…
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 42A82
Mathematics Subject Classification ID: 44A60
zbMATH DE Number: 1943365
zbMATH Keywords: countable abelian semigroup with zero
zbMATH Keywords: positive definite function
zbMATH Keywords: character
zbMATH Keywords: moment function
zbMATH Keywords: semiperfect semigroup
zbMATH Keywords: determinate moment sequence
zbMATH Keywords: indeterminate moment sequence
zbMATH Keywords: determinate measure
zbMATH Keywords: indeterminate measure
zbMATH Keywords: \(N\)-extremal measure
reviewed by: Alexander Tovstolis
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1943365