Weakly Stationary Random Fields, Invariant Subspaces and Applications: 1st Edition (Hardback) book cover

Weakly Stationary Random Fields, Invariant Subspaces and Applications

1st Edition

By Vidyadhar S. Mandrekar, David A. Redett

Chapman and Hall/CRC

182 pages

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pub: 2017-11-06
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Description

The first book to examine weakly stationary random fields and their connections with invariant subspaces (an area associated with functional analysis). It reviews current literature, presents central issues and most important results within the area. For advanced Ph.D. students, researchers, especially those conducting research on Gaussian theory.

Table of Contents

Introduction

Weakly Stationary Sequences

Preliminaries

Examples of Weakly Stationary Sequences

Spectral Representation of the Covariance Function

Examples of Spectral Measures

Canonical Isomorphism between L2(T; FX) and L(X)

Spectral Representation of a Weakly Stationary Sequence

The Shift Operator on L(X)

Moving Averages and Densities

Regular and Singular Sequences

Examples of Regular and Singular Sequences

The Wold Decomposition

The Theory of Regular Sequences

The Concordance Theorem

Maximal Factors and Innovation Processes

The Szego-Krein-Kolmogorov Theorem

Weakly Stationary Random Fields

Preliminaries

Examples

Random Field with Discrete Spectral Measure

Product Random Field

White Noise Random Field

Moving Average Random Field

Regularity and Singularity

Examples

Horizontally and Vertically Singular

Horizontally Regular and Vertically Singular

Horizontally and Vertically Regular

Horizontal and Vertical Wold Decomposition

Regularity and the Spectral Measure

Spectral Measures and Spectral-type Wold Decompositions

The Fourfold Wold Decomposition

Quarter-plane Moving Average Representations

Helson-Lowdenslager Theory

Semi-group Moving Average Representations

Wold-Type Decompositions

Invariant Subspaces

The Halmos Decomposition

Invariant Subspaces of L2(T)

Invariant Subspaces of H2(T)

The Halmos Fourfold Decomposition

The Doubly Commuting Condition

Invariant Subspaces of L2(T2)

Invariant Subspaces of H2(T2)

Applications and Generalizations

Texture Identi_cation

Invariant Subspaces of Lp(T)

Harmonizable S_S Sequences

Invariant Subspaces of Lp(T2)

Harmonizable S_S Fields

Proper MA Representations and Outer Functions

Background Material

Projections

Orthogonally Scattered Set Functions

Representation Theorems

Bibliography

Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis
MAT037000
MATHEMATICS / Functional Analysis