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Marinus A. Kaashoek

Bio: Marinus A. Kaashoek is an academic researcher from VU University Amsterdam. The author has contributed to research in topics: Factorization & Operator theory. The author has an hindex of 37, co-authored 258 publications receiving 6498 citations. Previous affiliations of Marinus A. Kaashoek include Tel Aviv University & Centrum Wiskunde & Informatica.


Papers
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Book
01 Jan 1990
TL;DR: In this article, a triangle representation of a RKHS operator is used to define a block Toeplitz operator defined by a rational matrix function, where the matrix functions are defined by piecewise continuous functions.
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

1,012 citations

Book
01 Jan 1966
TL;DR: In this article, the authors give an introduction to the theory of unbounded linear operators between Banach spaces, and the important notions of closed and closable operators and their conjugates with much attention paid to ordinary and partial differential 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.

519 citations


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01 Nov 1981
TL;DR: In this paper, the authors studied the effect of local derivatives on the detection of intensity edges in images, where the local difference of intensities is computed for each pixel in the image.
Abstract: Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

1,829 citations

Journal ArticleDOI
TL;DR: In this article, a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed, and concepts of controllability and observability are defined for the impulsive modes.
Abstract: Systems of the form E\dot{x}=Ax + Bu, y=Cx , with E singular, are studied. Of particular interest are the impulsive modes that may appear in the free-response of such systems when arbitrary initial conditions are permitted, modes that are associated with natural system frequencies at infinity. A generalized definition of system order that incorporates these impulsive degrees of freedom is proposed, and concepts of controllability and observability are defined for the impulsive modes. Allowable equivalence transformations of such singular systems are specified. The present framework is shown to overcome several difficulties inherent in other treatments of singular systems, and to extend, in a natural and satisfying way, many results previously known only for regular state-space systems.

1,042 citations

Journal ArticleDOI
TL;DR: A lifting technique is used to describe the continuous-time (i.e., intersample) behavior of sampled-data systems, and to obtain a complete solution to the problem of parameterizing all controllers that constrain the L/sup 2/-induced norm of a sampled- data system to within a certain bound.
Abstract: The authors present a framework for dealing with continuous-time periodic systems. The main tool is a lifting technique which provides a strong correspondence between continuous-time periodic systems and certain types of discrete-time time-invariant systems with infinite-dimensional input and output spaces. Despite the infinite dimensionality of the input and output spaces, a lifted system has a finite-dimensional state space if the original system does. This fact permits rather constructive methods for analyzing these systems. As a demonstration of the utility of this framework, the authors use it to describe the continuous-time (i.e., intersample) behavior of sampled-data systems, and to obtain a complete solution to the problem of parameterizing all controllers that constrain the L/sup 2/-induced norm of a sampled-data system to within a certain bound. >

650 citations