Weakly Stationary Random Fields, Invariant Subspaces and Applications

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