| Preface | p. vii |
| Introduction | p. 1 |
| Formulation of Basic Results | p. 9 |
| Statement of the problem | p. 9 |
| Formulation of the results (multidimensional case) | p. 14 |
| Basic results | p. 14 |
| Generalizations | p. 17 |
| Formulation of the results (one-dimensional case) | p. 18 |
| Basic results for the scalar equation | p. 19 |
| Vector equations | p. 22 |
| Examples of kernels of class R and solutions to the basic equation | p. 25 |
| Formula for the error of the optimal estimate | p. 29 |
| Numerical Solution of the Basic Integral Equation in Distributions | p. 33 |
| Basic ideas | p. 33 |
| Theoretical approaches | p. 37 |
| Multidimensional equation | p. 43 |
| Numerical solution based on the approximation of the kernel | p. 46 |
| Asymptotic behavior of the optimal filter as the white noise component goes to zero | p. 54 |
| A general approach | p. 57 |
| Proofs | p. 65 |
| Proof of Theorem 2.1 | p. 65 |
| Proof of Theorem 2.2 | p. 73 |
| Proof of Theorems 2.4 and 2.5 | p. 79 |
| Another approach | p. 84 |
| Singular Perturbation Theory for a Class of Fredholm Integral Equations Arising in Random Fields Estimation Theory | p. 87 |
| Introduction | p. 87 |
| Auxiliary results | p. 90 |
| Asymptotics in the case n = 1 | p. 93 |
| Examples of asymptotical solutions: case n = 1 | p. 98 |
| Asymptotics in the case n > 1 | p. 103 |
| Examples of asymptotical solutions: case n > 1 | p. 105 |
| Estimation and Scattering Theory | p. 111 |
| The direct scattering problem | p. 111 |
| The direct scattering problem | p. 111 |
| Properties of the scattering solution | p. 114 |
| Properties of the scattering amplitude | p. 120 |
| Analyticity in k of the scattering solution | p. 121 |
| High-frequency behavior of the scattering solutions | p. 123 |
| Fundamental relation between u[superscript +] and u[superscript -] | p. 127 |
| Formula for det S(k) and state the Levinson Theorem | p. 128 |
| Completeness properties of the scattering solutions | p. 131 |
| Inverse scattering problems | p. 134 |
| Inverse scattering problems | p. 134 |
| Uniqueness theorem for the inverse scattering problem | p. 134 |
| Necessary conditions for a function to be a scattering amplitude | p. 135 |
| A Marchenko equation (M equation) | p. 136 |
| Characterization of the scattering data in the 3D inverse scattering problem | p. 138 |
| The Born inversion | p. 141 |
| Estimation theory and inverse scattering in R[superscript 3] | p. 150 |
| Applications | p. 159 |
| What is the optimal size of the domain on which the data are to be collected? | p. 159 |
| Discrimination of random fields against noisy background | p. 161 |
| Quasioptimal estimates of derivatives of random functions | p. 169 |
| Introduction | p. 169 |
| Estimates of the derivatives | p. 170 |
| Derivatives of random functions | p. 172 |
| Finding critical points | p. 180 |
| Derivatives of random fields | p. 181 |
| Stable summation of orthogonal series and integrals with randomly perturbed coefficients | p. 182 |
| Introduction | p. 182 |
| Stable summation of series | p. 184 |
| Method of multipliers | p. 185 |
| Resolution ability of linear systems | p. 185 |
| Introduction | p. 185 |
| Resolution ability of linear systems | p. 187 |
| Optimization of resolution ability | p. 191 |
| A general definition of resolution ability | p. 196 |
| Ill-posed problems and estimation theory | p. 198 |
| Introduction | p. 198 |
| Stable solution of ill-posed problems | p. 205 |
| Equations with random noise | p. 216 |
| A remark on nonlinear (polynomial) estimates | p. 230 |
| Auxiliary Results | p. 233 |
| Sobolev spaces and distributions | p. 233 |
| A general imbedding theorem | p. 233 |
| Sobolev spaces with negative indices | p. 236 |
| Eigenfunction expansions for elliptic selfadjoint operators | p. 241 |
| Resoluion of the identity and integral representation of selfadjoint operators | p. 241 |
| Differentiation of operator measures | p. 242 |
| Carleman operators | p. 246 |
| Elements of the spectral theory of elliptic operators in L[superscript 2](R[superscript r]) | p. 249 |
| Asymptotics of the spectrum of linear operators | p. 260 |
| Compact operators | p. 260 |
| Basic definitions | p. 260 |
| Minimax principles and estimates of eigenvalues and singular values | p. 262 |
| Perturbations preserving asymptotics of the spectrum of compact operators | p. 265 |
| Statement of the problem | p. 265 |
| A characterization of the class of linear compact operators | p. 266 |
| Asymptotic equivalence of s-values of two operators | p. 268 |
| Estimate of the remainder | p. 270 |
| Unbounded operators | p. 274 |
| Asymptotics of eigenvalues | p. 275 |
| Asymptotics of eigenvalues (continuation) | p. 283 |
| Asymptotics of s-values | p. 284 |
| Asymptotics of the spectrum for quadratic forms | p. 287 |
| Proof of Theorem 2.3 | p. 293 |
| Trace class and Hilbert-Schmidt operators | p. 297 |
| Trace class operators | p. 297 |
| Hilbert-Schmidt operators | p. 298 |
| Determinants of operators | p. 299 |
| Elements of probability theory | p. 300 |
| The probability space and basic definitions | p. 300 |
| Hilbert space theory | p. 306 |
| Estimation in Hilbert space L[superscript 2]([Omega], U, P) | p. 310 |
| Homogeneous and isotropic random fields | p. 312 |
| Estimation of parameters | p. 315 |
| Discrimination between hypotheses | p. 317 |
| Generalized random fields | p. 319 |
| Kalman filters | p. 320 |
| Analytical Solution of the Basic Integral Equation for a Class of One-Dimensional Problems | p. 325 |
| Introduction | p. 326 |
| Proofs | p. 329 |
| Integral Operators Basic in Random Fields Estimation Theory | p. 337 |
| Introduction | p. 337 |
| Reduction of the basic integral equation to a boundary-value problem | p. 341 |
| Isomorphism property | p. 349 |
| Auxiliary material | p. 354 |
| Bibliographical Notes | p. 359 |
| Bibliography | p. 363 |
| Symbols | p. 371 |
| Index | p. 373 |
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