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Preface

Contents

Overview / 1：

Methods of Operator Approximation in System Modelling / I：

Nonlinear Operator Approximation with Preassigned Accuracy / 2：

Introduction / 2.1：

Generic formulation of the problem / 2.2：

Operator approximation in space C([0; 1]) / 2.3：

Operator approximation in Banach spaces by polynomial operators / 2.4：

Approximation on compact sets in topological vector spaces / 2.5：

Approximation on noncompact sets in Hilbert spaces / 2.6：

Special results for maps into Banach spaces / 2.7：

Concluding remarks / 2.8：

Interpolation of Nonlinear Operators 65 / 3：

Lagrange interpolation in Banach spaces / 3.1：

Weak interpolation of nonlinear operators / 3.3：

Some related results / 3.4：

Realistic Operators and their Approximation / 3.5：

Formalization of concepts related to description of real-world objects / 4.1：

Approximation of RÂícontinuous operators / 4.3：

Methods of Best Approximation for Nonlinear Operators / 4.4：

Best Approximation of nonlinear operators in Banach spaces: Deterministic case / 5.1：

Estimation of mean and covariance matrix for random vectors / 5.3：

Best Hadamard-quadratic approximation / 5.4：

Best polynomial approximation / 5.5：

Best causal approximation / 5.6：

Best hybrid approximations / 5.7：

Optimal Estimation of Random Vectors / 5.8：

Computational Methods for Optimal Filtering of Stochastic Signals / 6：

Optimal linear Filtering in Finite dimensional vector spaces / 6.1：

Optimal linear Filtering in Hilbert spaces / 6.3：

Optimal causal linear Filtering with piecewise constant memory / 6.4：

Optimal causal polynomial Filtering with arbitrarily variable memory / 6.5：

Optimal nonlinear Filtering with no memory constraint / 6.6：

Computational Methods for Optimal Compression and Reconstruction of Random Data / 6.7：

Standard Principal Component Analysis and Karhunen-Loeeve transform (PCA{KLT) / 7.1：

Rank-constrained matrix approximations / 7.3：

Generic PCA{KLT / 7.4：

Optimal hybrid transform based on Hadamard-quadratic approximation / 7.5：

Optimal transform formed by a combination of nonlinear operators / 7.6：

Optimal generalized hybrid transform / 7.7：

Bibliography / 7.8：

Index

Preface

Contents

Overview / 1：

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The Simplest Problem in Calculus of Variations / 1：

The Optimal Control Problem / 2：

Existence and Continuity Properties of Optimal Controls / 3：

Dynamic Programming / 4：

Stochastic Differential Equations and Markov Diffusion Processes / 5：

Optimal Control of Markov Diffusion Processes / 6：

Appendices

The Simplest Problem in Calculus of Variations / 1：

The Optimal Control Problem / 2：

Existence and Continuity Properties of Optimal Controls / 3：

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