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.
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 Szeg o-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
Vidyadhar Mandrekar and David A. Reddett