Preface -- -- Contents -- -- 1 Overview -- -- I Methods of Operator Approximation in System Modelling -- -- 2 Nonlinear Operator Approximation with Preassigned Accuracy -- -- 2.1 Introduction -- -- 2.2 Generic formulation of the problem -- -- 2.3 Operator approximation in space C([0; 1]): -- -- 2.4 Operator approximation in Banach spaces by polynomial operators -- -- 2.5 Approximation on compact sets in topological vector spaces -- -- 2.6 Approximation on noncompact sets in Hilbert spaces -- -- 2.7 Special results for maps into Banach spaces -- -- 2.8 Concluding remarks -- -- 3 Interpolation of Nonlinear Operators 65 -- -- 3.1 Introduction -- -- 3.2 Lagrange interpolation in Banach spaces -- -- 3.3 Weak interpolation of nonlinear operators -- -- 3.4 Some related results -- -- 3.5 Concluding remarks -- -- 4 Realistic Operators and their Approximation -- -- 4.1 Introduction -- -- 4.2 Formalization of concepts related to description of real-world objects -- -- 4.3 Approximation of R¡continuous operators -- -- 4.4 Concluding remarks -- -- 5 Methods of Best Approximation for Nonlinear Operators -- -- 5.1 Introduction -- -- 5.2 Best Approximation of nonlinear operators in Banach spaces: Deterministic case -- -- 5.3 Estimation of mean and covariance matrix for random vectors -- -- 5.4 Best Hadamard-quadratic approximation -- -- 5.5 Best polynomial approximation -- -- 5.6 Best causal approximation -- -- 5.7 Best hybrid approximations -- -- 5.8 Concluding remarks -- -- II Optimal Estimation of Random Vectors -- -- 6 Computational Methods for Optimal Filtering of Stochastic Signals -- -- 6.1 Introduction -- -- 6.2 Optimal linear Filtering in Finite dimensional vector spaces -- -- 6.3 Optimal linear Filtering in Hilbert spaces -- -- 6.4 Optimal causal linear Filtering with piecewise constant memory -- -- 6.5 Optimal causal polynomial Filtering with arbitrarily variable memory -- -- 6.6 Optimal nonlinear Filtering with no memory constraint -- -- 6.7 Concluding remarks -- -- 7 Computational Methods for Optimal Compression and -- Reconstruction of Random Data -- -- 7.1 Introduction -- -- 7.2 Standard Principal Component Analysis and Karhunen-Loeeve transform (PCA{KLT) -- -- 7.3 Rank-constrained matrix approximations -- -- 7.4 Generic PCA{KLT -- -- 7.5 Optimal hybrid transform based on Hadamard-quadratic approximation -- -- 7.6 Optimal transform formed by a combination of nonlinear operators -- -- 7.7 Optimal generalized hybrid transform -- -- 7.8 Concluding remarks -- -- Bibliography -- -- Index<P>.