| Introduction |
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ix | |
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1 Weakly Stationary Sequences |
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1 | (40) |
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1 | (1) |
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1.2 Examples of Weakly Stationary Sequences |
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2 | (1) |
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1.3 Spectral Representation of the Covariance Function |
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2 | (1) |
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1.4 Examples of Spectral Measures |
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3 | (1) |
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1.5 Canonical Isomorphism between L2(T, Fx) and L(X) |
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4 | (1) |
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1.6 Spectral Representation of a Weakly Stationary Sequence |
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5 | (1) |
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1.7 The Shift Operator on L(X) |
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6 | (2) |
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1.8 Moving Averages and Densities |
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8 | (7) |
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1.9 Regular and Singular Sequences |
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15 | (1) |
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1.10 Examples of Regular and Singular Sequences |
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15 | (3) |
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1.11 The Wold Decomposition |
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18 | (4) |
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1.12 The Theory of Regular Sequences |
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22 | (1) |
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1.13 The Concordance Theorem |
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23 | (3) |
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1.14 Maximal Factors and Innovation Processes |
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26 | (4) |
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1.15 The Szego-Krein-Kolmogorov Theorem |
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30 | (9) |
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1.16 Remarks and Related Literature |
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39 | (2) |
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2 Weakly Stationary Random Fields |
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41 | (68) |
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41 | (3) |
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44 | (4) |
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2.2.1 Random Field with Discrete Spectral Measure |
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45 | (1) |
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2.2.2 Product Random Field |
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45 | (1) |
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2.2.3 White Noise Random Field |
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45 | (1) |
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2.2.4 Moving Average Random Field |
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45 | (3) |
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2.3 Regularity and Singularity |
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48 | (1) |
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49 | (2) |
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2.4.1 Horizontally and Vertically Singular |
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49 | (1) |
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2.4.2 Horizontally Regular and Vertically Singular |
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49 | (1) |
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2.4.3 Horizontally and Vertically Regular |
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50 | (1) |
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2.5 Horizontal and Vertical Wold Decomposition |
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51 | (4) |
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2.6 Regularity and the Spectral Measure |
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55 | (5) |
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2.7 Spectral Measures and Spectral-type Wold Decompositions |
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60 | (10) |
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2.8 The Fourfold Wold Decomposition |
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70 | (8) |
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2.9 Quarter-plane Moving Average Representations |
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78 | (6) |
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2.10 Helson-Lowdenslager Theory |
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84 | (13) |
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2.11 Semigroup Moving Average Representations |
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97 | (4) |
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2.12 Wold-Type Decompositions |
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101 | (5) |
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2.13 Remarks and Related Literature |
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106 | (3) |
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109 | (26) |
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3.1 The Halmos Decomposition |
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109 | (4) |
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3.2 Invariant Subspaces of L2(T) |
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113 | (2) |
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3.3 Invariant Subspaces of H2(T) |
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115 | (1) |
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3.4 The Halmos Fourfold Decomposition |
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115 | (9) |
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3.5 The Doubly Commuting Condition |
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124 | (2) |
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3.6 Invariant Subspaces of L2(T2) |
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126 | (6) |
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3.7 Invariant Subspaces of H2(T2) |
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132 | (1) |
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3.8 Remarks and Related Literature |
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133 | (2) |
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4 Applications and Generalizations |
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135 | (36) |
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4.1 Texture Identification |
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135 | (13) |
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4.2 Invariant Subspaces of LP(T) |
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148 | (3) |
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4.3 Harmonizable SαS Sequences |
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151 | (7) |
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4.4 Invariant Subspaces of LP(T2) |
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158 | (5) |
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4.5 Harmonizable SαS Fields |
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163 | (3) |
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4.6 Proper MA Representations and Outer Functions |
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166 | (3) |
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4.7 Remarks and Related Literature |
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169 | (2) |
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171 | (6) |
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171 | (1) |
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A.2 Orthogonally Scattered Set Functions |
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172 | (1) |
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A.3 Representation Theorems |
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173 | (4) |
| Bibliography |
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177 | (4) |
| Index |
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181 | |
