Seymour Goldberg

Bio: Seymour Goldberg is an academic researcher from University of Maryland, College Park. The author has contributed to research in topics: Operator theory & Hilbert space. The author has an hindex of 13, co-authored 41 publications receiving 3146 citations. Previous affiliations of Seymour Goldberg include Silver Spring Networks & Tel Aviv University.

Classes of Linear Operators

Abstract: Preface to Volume II Table of contents of Volume II Introduction PART V: TRIANGULAR REPRESENTATIONS XX Additive lower-upper triangular decompositions of operators 1 Additive lower-upper triangular decompositions relative to finite chains 2 Preliminaries about chains 3 Diagonals 4 Chains on Hilbert space 5 Triangular algebras 6 Riemann-Stieltjes integration along chains 7 Additive lower-upper decomposition theorem 8 Additive lower-upper decomposition of a Hilbert-Schmidt operator 9 Multiplicative integration along chains 10 Basic properties of reproducing kernel Hilbert spaces and chains 11 Example of an additive LU-decomposition in a RKHS XXI Operators in triangular form 1 Triangular representation 2 Intermezzo about completely nonselfadjoint operators 3 Volterra operators with a one-dimensional imaginary part 4 Unicellular operators XXII Multiplicative lower-upper triangular decompositions of operators 1 LU-factorization with respect to a finite chain 2 The LU-factorization theorem 3 LU-factorizations of compact perturbations of the identity 4 LU-factorizatioris of Hilbert-Schmidt perturbations of the identity 5 LU-factorizations of integral operators 6 Triangular representations of operators close to unitary 7 LU-factorization of semi-separable integral operators 8 Generalised Wiener-Hopf equations 9 Generalised LU-factorization relative to discrete chains Comments on Part V Exercises to Part V PART VI: CLASSES OF TOEPLITZ OPERATORS XXIII Block Toeplitz operators 1 Preliminaries 2 Block Laurent operators 3 Block Toeplitz operators 4 Block Toeplitz operators defined by continuous functions 5 The Fredholm index of a block Toeplitz operator defined by a continuous function XXIV Toeplitz operators defined by rational matrix functions 1 Preliminaries 2 Invertibility and Fredholm index (scalar case) 3 Wiener-Hopf factorization 4 Invertibility and Fredholm index (matrix case) 5 Intermezzo about realisation 6 Inversion of a block Laurent operator 7 Explicit canonical factorization 8 Explicit inversion formulas 9 Explicit formulas for Fredholm characteristics 10 An example 11 Asymptotic formulas for determinants of block Toeplitz matrices XXV Toeplitz operators defined by piecewise continuous matrix functions 1 Piecewise continuous functions 2 Symbol and Fredholm index (scalar case) 3 Symbol and Fredholm index (matrix case) 4 Sums of products of Toeplitz operators defined by piecewise continuous functions 5 Sums of products of block Toeplitz operators defined by piecewise continuous functions Comments on Part VI Exercises to Part VI PART VII: CONTRACTIVE OPERATORS AND CHARACTERISTIC OPERATOR FUNCTIONS XXVI Block shift operators 1 Forward shifts and isometries 2 Parts of block shift operators 3 Invariant subspaces of forward shift operators XXVII Dilation theory 1 Preliminaries about contractions 2 Preliminaries about dilations 3 Isometric dilations 4 Unitary dilations

Unbounded Linear Operators

Abstract: This chapter gives an introduction to the theory of unbounded linear operators between Banach spaces. The important notions of closed and closable operators and their conjugates are analyzed with much attention paid to ordinary and partial differential operators. In particular, maximal and minimal operators and the properties of their inverses are studied. The chapter is divided into 6 sections. The first two sections are devoted to the general theory, and the other four sections deal mainly with differential operators.

Traces and determinants of linear operators

Abstract: A general theory of tracestr D A and determinantsdet D (I+A) in normed algebrasD of operators acting in Banach spacesB is proposed. In this approach trace and determinant are defined as continuous extensions of the corresponding functionals from finite dimensional operators. We characterize the algebras for which such extensions exist and describe sets of possible values of traces and determinants for the same operator in different algebras. In spite of the fact that the extended traces and determinants may differ in different algebrasD, operatorI+A (withA ∈D) is invertible inB if and only ifdet D (I+A) does not vanish. Cramer's rule and formulas for the resolvent are obtained and they are expressed in different algebras by the same formulas viadet D (I+A) andtr D (A). A large set of examples and illustrations are also presented.

Essentials of Robust Control

Abstract: 1. Introduction. 2. Linear Algebra. 3. Linear Systems. 4. H2 and Ha Spaces. 5. Internal Stability. 6. Performance Specifications and Limitations. 7. Balanced Model Reduction. 8. Uncertainty and Robustness. 9. Linear Fractional Transformation. 10. m and m- Synthesis. 11. Controller Parameterization. 12. Algebraic Riccati Equations. 13. H2 Optimal Control. 14. Ha Control. 15. Controller Reduction. 16. Ha Loop Shaping. 17. Gap Metric and ...u- Gap Metric. 18. Miscellaneous Topics. Bibliography. Index.

Wavelets and Subband Coding

Abstract: First published in 1995, Wavelets and Subband Coding offered a unified view of the exciting field of wavelets and their discrete-time cousins, filter banks, or subband coding. The book developed the theory in both continuous and discrete time, and presented important applications. During the past decade, it filled a useful need in explaining a new view of signal processing based on flexible time-frequency analysis and its applications. Since 2007, the authors now retain the copyright and allow open access to the book.

A fixed point theorem in partially ordered sets and some applications to matrix equations

Abstract: An analogue of Banach's fixed point theorem in partially ordered sets is proved in this paper, and several applications to linear and nonlinear matrix equations are discussed.

Continuous and discrete wavelet transforms

Abstract: This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ in terms of coherent states. Two types of coherent states are considered: Weyl–Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, called ’wavelets,’ which arise as translations and dilations of a single function. In each case it is shown how to represent any function in $L^2 ({\bf R})$ as a sum or integral of these states. Most of the paper is a survey of literature, most notably the work of I. Daubechies, A. Grossmann, and J. Morlet. A few results of the authors are included.