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Publishers' Page February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book Publishers' page i February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book Publishers' page ii February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book Publishers' page iii February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book Publishers' page iv February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book To the memory of my parents v February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book vi February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book Preface This book presents analytic theory of random fields estimation optimal by the criterion of minimum of the variance of the error of the estimate. This theory is a generalization of the classical Wiener theory. Wiener's theory has been developed for optimal estimation of stationary random processes, that is, random functions of one variable. Random fields are random func- tions of several variables. Wiener's theory was based on the analytical solution of the basic integral equation of estimation theory. This equation for estimation of stationary random processes was Wiener-Hopf-type of equation, originally on a positive semiaxis. About 25 years later the theory of such equations has been developed for the case of finite intervals. The assumption of stationarity of the processes was vital for the theory. An- alytical formulas for optimal estimates (filters) have been obtained under the assumption that the spectral density of the stationary process is a pos- itive rational function. We generalize Wiener's theory in several directions. First, estimation theory of random fields and not only random processes is developed. Secondly, the stationarity assumption is dropped. Thirdly, the assumption about rational spectral density is generalized in this book: we consider kernels of positive rational functions of arbitrary elliptic self- adjoint operators on the whole space. The domain of observation of the signal does not enter into the definition of the kernel. These kernels are correlation functions of random fields and therefore the class of such kernels defines the class of random fields for which analytical estimation theory is developed. In the appendix we consider even more general class of kernels, namely kernels R(x; y), which solve the equation QR = P δ(x y). Here − P and Q are elliptic operators, and δ(x y) is the delta-function. We − study singular perturbation problem for the basic integral equation of esti- mation theory Rh = f. The solution to this equation, which is of interest vii February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book viii Random Fields Estimation Theory in estimation theory, is a distribution, in general. The perturbed equation, 2 h +Rh = f has the unique solution in L (D). The singular perturbation problem consists of the study of the asymptotics of h as 0. This theory ! is not only of mathematical interest, but also a basis for the numerical solu- tion of the basic integral equation in distributions. We discuss the relation between estimation theory and quantum-mechanical non-relativistic scat- tering theory. Applications of the estimation theory are also discussed. The presentation in this book is based partly on the author's earlier monographs [Ramm (1990)] and [Ramm (1996)], but also contains recent results [Ramm (2002)], [Ramm (2003)],[Kozhevnikov and Ramm (2005)], and [Ramm and Shifrin (2005)]. The book is intended for researchers in probability and statistics, anal- ysis, numerical analysis, signal estimation and image processing, theoreti- cally inclined electrical engineers, geophysicists, and graduate students in these areas. Parts of the book can be used in graduate courses in proba- bilty and statistics. The analytical tools that the author uses are not usual for statistics and probability. These tools include spectral theory of elliptic operators, pseudodifferential operators, and operator theory. The presen- tation in this book is essentially self-contained. Auxiliary material which we use is collected in Chapter 8. February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book Contents Preface vii 1. Introduction 1 2. Formulation of Basic Results 9 2.1 Statement of the problem . 9 2.2 Formulation of the results (multidimensional case) . 14 2.2.1 Basic results . 14 2.2.2 Generalizations . 17 2.3 Formulation of the results (one-dimensional case) . 18 2.3.1 Basic results for the scalar equation . 19 2.3.2 Vector equations . 22 2.4 Examples of kernels of class and solutions to the basic R equation . 25 2.5 Formula for the error of the optimal estimate . 29 3. Numerical Solution of the Basic Integral Equation in Dis- tributions 33 3.1 Basic ideas . 33 3.2 Theoretical approaches . 37 3.3 Multidimensional equation . 43 3.4 Numerical solution based on the approximation of the kernel 46 3.5 Asymptotic behavior of the optimal filter as the white noise component goes to zero . 54 3.6 A general approach . 57 ix February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book x Random Fields Estimation Theory 4. Proofs 65 4.1 Proof of Theorem 2.1 . 65 4.2 Proof of Theorem 2.2 . 73 4.3 Proof of Theorems 2.4 and 2.5 . 79 4.4 Another approach . 84 5. Singular Perturbation Theory for a Class of Fredholm Integral Equations Arising in Random Fields Estimation Theory 87 5.1 Introduction . 87 5.2 Auxiliary results . 90 5.3 Asymptotics in the case n = 1 . 93 5.4 Examples of asymptotical solutions: case n = 1 . 98 5.5 Asymptotics in the case n > 1 . 103 5.6 Examples of asymptotical solutions: case n > 1 . 105 6. Estimation and Scattering Theory 111 6.1 The direct scattering problem . 111 6.1.1 The direct scattering problem . 111 6.1.2 Properties of the scattering solution . 114 6.1.3 Properties of the scattering amplitude . 120 6.1.4 Analyticity in k of the scattering solution . 121 6.1.5 High-frequency behavior of the scattering solutions . 123 + 6.1.6 Fundamental relation between u and u− . 127 6.1.7 Formula for det S(k) and the Levinson Theorem . 128 6.1.8 Completeness properties of the scattering solutions . 131 6.2 Inverse scattering problems . 134 6.2.1 Inverse scattering problems . 134 6.2.2 Uniqueness theorem for the inverse scattering problem 134 6.2.3 Necessary conditions for a function to be a scatterng amplitude . 135 6.2.4 A Marchenko equation (M equation) . 136 6.2.5 Characterization of the scattering data in the 3D in- verse scattering problem . 138 6.2.6 The Born inversion . 141 6.3 Estimation theory and inverse scattering in R3 . 150 7. Applications 159 February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book Contents xi 7.1 What is the optimal size of the domain on which the data are to be collected? . 159 7.2 Discrimination of random fields against noisy background . 161 7.3 Quasioptimal estimates of derivatives of random functions . 169 7.3.1 Introduction . 169 7.3.2 Estimates of the derivatives . 170 7.3.3 Derivatives of random functions . 172 7.3.4 Finding critical points . 180 7.3.5 Derivatives of random fields . 181 7.4 Stable summation of orthogonal series and integrals with randomly perturbed coefficients . 182 7.4.1 Introduction . 182 7.4.2 Stable summation of series . 184 7.4.3 Method of multipliers . 185 7.5 Resolution ability of linear systems . 185 7.5.1 Introduction . 185 7.5.2 Resolution ability of linear systems . 187 7.5.3 Optimization of resolution ability . 191 7.5.4 A general definition of resolution ability . 196 7.6 Ill-posed problems and estimation theory . 198 7.6.1 Introduction . 198 7.6.2 Stable solution of ill-posed problems . 205 7.6.3 Equations with random noise . 216 7.7 A remark on nonlinear (polynomial) estimates . 230 8. Auxiliary Results 233 8.1 Sobolev spaces and distributions . 233 8.1.1 A general imbedding theorem . 233 8.1.2 Sobolev spaces with negative indices . 236 8.2 Eigenfunction expansions for elliptic selfadjoint operators . 241 8.2.1 Resoluion of the identity and integral representation of selfadjoint operators . 241 8.2.2 Differentiation of operator measures . 242 8.2.3 Carleman operators . 246 8.2.4 Elements of the spectral theory of elliptic operators in L2(Rr) . 249 8.3 Asymptotics of the spectrum of linear operators . 260 8.3.1 Compact operators . 260 8.3.1.1 Basic definitions . 260 February 12, 2006 10:52 WSPC/Book Trim Size for 9in x 6in book xii Random Fields Estimation Theory 8.3.1.2 Minimax principles and estimates of eigen- values and singular values . 262 8.3.2 Perturbations preserving asymptotics of the spectrum of compact operators . 265 8.3.2.1 Statement of the problem . 265 8.3.2.2 A characterization of the class of linear com- pact operators . 266 8.3.2.3 Asymptotic equivalence of s-values of two op- erators . 268 8.3.2.4 Estimate of the remainder . 270 8.3.2.5 Unbounded operators . 274 8.3.2.6 Asymptotics of eigenvalues . 275 8.3.2.7 Asymptotics of eigenvalues (continuation) . 283 8.3.2.8 Asymptotics of s-values . 284 8.3.2.9 Asymptotics of the spectrum for quadratic forms . 287 8.3.2.10 Proof of Theorem 2.3 . 293 8.3.3 Trace class and Hilbert-Schmidt operators .
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