Stationary processes on the dyadic tree: prediction and covariance extension
type: Item
instance of: scholarly article
title: Stationary processes on the dyadic tree: prediction and covariance extension
zbMATH Open document ID: 1103.60037
author: Daniel Alpay
author: Dan Volok
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2006-03-20T00:00:00Z
review text: The authors are interested in the study of linear systems indexed by the nodes of an infinite homogeneous tree of order 2. In their previous two papers [Integral Equations Oper.\ Theory 53, No.~1, 1--22 (2005; Zbl 1100.93010); C.~R., Math., Acad.\ Sci.\ Paris 336, No.~4, 293--298 (2003; Zbl 1030.46023)], they studied the stationary and nonstationary case and defined a corresponding function theory in each of these cases. In this third paper of the study, the case of stationary and isotropic stochastic processes of order \(q=2\) is considered. In this way, some necessary definitions such as point from the infinity, order on the tree, stationarity and prediction, and analogues of the Yule-Walker equations are introduced. The analogue of the band extension when the integers are replaced by the nodes of the dyadic tree is studied. A theorem for the existence and unicity of the band extension as a positive and stationary matrix is proved. In the last section, the special case of an isotropic process is analysed, using the obtained band extension methods.
reviewed by: Ilie Valuşescu
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60G12
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 46N30
Mathematics Subject Classification ID: 47A48
Mathematics Subject Classification ID: 93B28
zbMATH DE Number: 5013374
zbMATH Keywords: isotropic
zbMATH Keywords: infinite homogeneous tree
zbMATH Keywords: nodes
zbMATH Keywords: Yule-Walker equations
zbMATH Keywords: Schur complement
zbMATH Keywords: band extension
MaRDI profile type: Publication
full work available at URL: j.crma.2005.12.018
OpenAlex ID: W1991518462
cites work: Point evaluation and Hardy space: the multiscale case
cites work: Interpolation et espace de Hardy sur l'arbre homogène dyadique : le cas stationnaire. (Interpolation and the Hardy space on the homogeneous dyadic tree: the stationary case)
cites work: Point evaluation and Hardy space on a homogeneous tree
cites work: Stationary processes indexed by a homogeneous tree
cites work: Multiscale system theory
cites work: Q4769263
cites work: Extensions of band matrices with band inverses
cites work: Classes of linear operators. Vol. II
DOI: 10.1016/J.CRMA.2005.12.018
description: scientific article; zbMATH DE number 5013374