Integral characterization of elementary definitizable functions
type: Item
instance of: scholarly article
title: Integral characterization of elementary definitizable functions
zbMATH Open document ID: 0734.43003
DOI: 10.1007/BF02570852
author: P. H. Maserick
author: El-Hassan Youssfi
published in: Mathematische Zeitschrift
publication date: 1992-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174380
review text: Let \(S\) be a commutative semigroup with involution and identity and let \(\Gamma\) be a subset of semicharacters on S equipped with the topology of pointwise convergence. A kernel \(Log_{\rho_ 0}(s,\rho)\) such that \(Log_{\rho_ 0}(s,\rho)\) is a branch of \(\log (\rho (s)/\rho_ 0(s))\) is introduced whose imaginary part agrees with the classical Lévy- kernel for groups. The exponential kernel \(Ex_{\kappa}\) is then defined as a truncation of the power series expansion of \(\exp (Log_{\rho_ 0}(s,\rho))\). An integral representation for a class of definitizable functions with one singularity \(\rho_ 0\) is derived. When \(S\) is a group with \(s^*=s^{-1}\) or more generally an inverse semigroup and \(\rho_ 0\equiv 1\), this representation reduces to the form \[ \phi(s)=\omega(s)+\int_{\Gamma}(\rho(s)-Ex_{\kappa}(s,\rho))d\mu(\rho),\quad (s\in S,\quad \rho \in \Gamma), \] where \(\mu\) is a non-negative measure and \(\omega\) satisfies an elementary functional equation. The above formula coincides with the celebrated Lévy-Khinchin formula for conditionally positive definite functions as well as the more recent extension of Sásvári for the group case. In this special case, the \(Log_{\rho_ 0}(s,\rho)\) reduces to a branch of \(\text{arg}(\rho(s))\) and the definitizable functions introduced here are a subclass of those studied by Berg and Sásvári. For an arbitrary semigroup \(S\), the integral representation is more complicated since the kernel may have a non-zero real part. Finally, we apply our results to solve Haviland's indefinite moment problem.
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 47B50
zbMATH DE Number: 4214679
zbMATH Keywords: commutative semigroup with involution
zbMATH Keywords: semicharacters
zbMATH Keywords: kernel
zbMATH Keywords: truncation
zbMATH Keywords: power series expansion
zbMATH Keywords: integral representation
zbMATH Keywords: definitizable functions
zbMATH Keywords: Lévy-Khinchin formula
zbMATH Keywords: conditionally positive definite functions
MaRDI profile type: Publication
cites work: Positive definite functions on Abelian semigroups
cites work: Q3755956
cites work: Exponentially bounded positive definite functions
cites work: Functions with a finite number of negative squares on semigroups
cites work: Les fonctions de Lévy existent ! (Lévy\(functions\) do exist !)
cites work: Q3848243
cites work: The L�vy-Hin?in representation of negative definite functions
cites work: The l 1 -Algebra of a Commutative Semigroup
cites work: BV-Functions, Positive-Definite Functions and Moment Problems
cites work: Moments of measures on convex bodies
cites work: A Levy-Khinchin formula for semigroups with involutions
cites work: Q4204600
cites work: Probability distributions on locally compact Abelian groups
cites work: Q3478990
description: scientific article; zbMATH DE number 4214679