Generalized Toeplitz kernels, stationarity and harmonizability
type: Item
instance of: scholarly article
title: Generalized Toeplitz kernels, stationarity and harmonizability
zbMATH Open document ID: 0579.42009
DOI: 10.1007/BF02790193
author: Mischa Cotlar
author: Cora Sadosky
published in: Journal d'Analyse Mathématique
publication date: 1985-01-01T00:00:00Z
review text: Given a Hilbert space \({\mathfrak M}\), an \({\mathfrak M}\)-process is a function X:\({\mathbb{Z}}\to {\mathfrak M}\) with its covariance kernel \(K(m,n)=(X(m),X(n))\). X is called generalized stationary process if its kernel K(m,n) is generalized Toeplitz kernel i.e. satisfies \(K(m,n)=K(m+1\), \(n+1)\) for \(m\neq -1\), \(n\neq -1\). An \({\mathfrak M}\)-valued process is said to be generalized harmonizable one if it admits representation \(X(n)=\hat V_ 1(n)\), \(n\geq 0\), \(X(n)=\hat V_ 2(n)\), \(n<0\) where \(V_ 1,V_ 2\) are bounded \({\mathfrak M}\)-valued measures on \({\mathbb{T}}\). The authors establish connection between these two notions and discuss some equivalent definitions of generalized harmonizable process. It is proved that for certain operations \(\Lambda\) the equation \(\Lambda X=Y\), Y being generalized stationary process, has a generalized stationary solution if and only if it has a generalized harmonizable one.
Mathematics Subject Classification ID: 42A82
Mathematics Subject Classification ID: 60G20
zbMATH DE Number: 3927591
zbMATH Keywords: stochastic process
zbMATH Keywords: Hilbert space
zbMATH Keywords: Toeplitz kernel
zbMATH Keywords: stationary process
MaRDI profile type: Publication
cites work: Q3957466
cites work: Q3238714
cites work: Q3894416
cites work: Q4169778
cites work: Q4124040
cites work: On orthogonally scattered dilations of bounded vector measures
cites work: Q4107698
cites work: On the linear prediction problem of certain non-stationary stochastic processes.
cites work: Dilation theorems for positive definite operator kernels having majorants
cites work: Q4064782
description: scientific article; zbMATH DE number 3927591