Random fields estimation / Alexander G. Ramm.

Includes bibliographical references (pages 363-369) and index.

Cover -- Preface -- Contents -- 1. Introduction -- 2. Formulation of Basic Results -- 2.1 Statement of the problem -- 2.2 Formulation of the results (multidimensional case) -- 2.2.1 Basic results -- 2.2.2 Generalizations -- 2.3 Formulation of the results (one-dimensional case) -- 2.3.1 Basic results for the scalar equation -- 2.3.2 Vector equations -- 2.4 Examples of kernels of class R and solutions to the basic equation -- 2.5 Formula for the error of the optimal estimate -- 3. Numerical Solution of the Basic Integral Equation in Distributions -- 3.1 Basic ideas -- 3.2 Theoretical approaches -- 3.3 Multidimensional equation -- 3.4 Numerical solution based on the approximation of the kernel -- 3.5 Asymptotic behavior of the optimal filter as the white noise component goes to zero -- 3.6 A general approach -- 4. Proofs -- 4.1 Proof of Theorem 2.1 -- 4.2 Proof of Theorem 2.2 -- 4.3 Proof of Theorems 2.4 and 2.5 -- 4.4 Another approach -- 5. Singular Perturbation Theory for a Class of Fredholm Integral Equations Arising in Random Fields Estimation Theory -- 5.1 Introduction -- 5.2 Auxiliary results -- 5.3 Asymptotics in the case n = 1 -- 5.4 Examples of asymptotical solutions: case n = 1 -- 5.5 Asymptotics in the case n> 1 -- 5.6 Examples of asymptotical solutions: case n> 1 -- 6. Estimation and Scattering Theory -- 6.1 The direct scattering problem -- 6.1.1 The direct scattering problem -- 6.1.2 Properties of the scattering solution -- 6.1.3 Properties of the scattering amplitude -- 6.1.4 Analyticity in k of the scattering solution -- 6.1.5 High-frequency behavior of the scattering solutions -- 6.1.6 Fundamental relation between u+ and u- -- 6.1.7 Formula for det S (k) and state the Levinson Theorem -- 6.1.8 Completeness properties of the scattering solutions -- 6.2 Inverse scattering problems -- 6.2.1 Inverse scattering problems -- 6.2.2 Uniqueness theorem for the inverse scattering problem -- 6.2.3 Necessary conditions for a function to be a scatterng amplitude -- 6.2.4 A Marchenko equation (M equation) -- 6.2.5 Characterization of the scattering data in the 3D inverse scattering probtem -- 6.2.6 The Born inversion -- 6.3 Estimation theory and inverse scattering in R3 -- 7. Applications -- 7.1 What is the optimal size of the domain on which the data are to be collected? -- 7.2 Discrimination of random fields against noisy background -- 7.3 Quasioptimal estimates of derivatives of random functions -- 7.3.1 Introduction -- 7.3.2 Estimates of the derivatives -- 7.3.3 Derivatives of random functions -- 7.3.4 Finding critical points -- 7.3.5 Derivatives of random fields -- 7.4 Stable summation of orthogonal series and integrals with randomly perturbed coefficients -- 7.4.1 Introduction -- 7.4.2 Stable summation of series -- 7.4.3 Method of multipliers -- 7.5 Resolution ability of linear systems -- 7.5.1 Introduction -- 7.5.2 Resolution ability of linear systems -- 7.5.3 Optimization of resolution ability -- 7.5.4 A general definition of resolution ability -- 7.6 Ill-posed problems and estimation theory -- 7.6.1 Introduction -- 7.6.2 Stable solution of ill-posed problems -- 7.6.3 Equations with random noise -- 7.7 A remark on nonlinear (polynomial) estimates -- 8. Auxiliary Results -- 8.1 Sobolev spaces and distributions -- 8.1.1 A general imbedding theorem -- 8.1.2 Sobolev space.

This book contains a novel theory of random fields estimation of Wiener type, developed originally by the author and presented here. No assumption about the Gaussian or Markovian nature of the fields are made. The theory, constructed entirely within the framework of covariance theory, is based on a detailed analytical study of a new class of multidimensional integral equations basic in estimation theory. This book is suitable for graduate courses in random fields estimation. It can also be used in courses in functional analysis, numerical analysis, integral equations, and scattering theory.

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