Current Works and Readings

Currently reading (11/24/2008 06:43 PM)

This is mainly a list of books and papers I am reading. Some of them are available online while others are not. I've also an del.icio.us account (see wiki for more info) where you can view some of the science pages I've bookmarked. Two Variable Polynomials acting as Lyapunov Functionals Jan C. Willems and Harry L. Trentelman, "On Quadratic Differential Forms," in {SIAM J. Control. Optim.}, 1998, pp. 1703-1749 (pdf) Making Tensor Products useful by canonical identifications with two variable polynomials Paul A. Fuhrmann and Uwe Helmke, "Tensored polynomial models ," in To appear in Giorgio Picci Festschrift, 2009 (dvi) Some geometric questions in the theory of linear systems Brockett Roger W., "Some geometric questions in the theory of linear systems," in IEEE Trans. Automatic Control, 1976, pp. 449--455 (info) Construction of Orthonormal Bases for H2 Brett Ninness and Fredrik Gustafsson, "A unifying Construction of Orthonormal Bases for System Identification," in Conference on Decision and Control, 1994, pp. 3388-3393 (info) Reproducing Kernels with Applications in Interpolation Baver Okutmustur, "Reproducing Kernel Hilbert Spaces," in Bilkent University, 2005 (info) Polynomial, Rational Models and More Paul A. Fuhrmann, "A Polynomial Approach to Linear Algebra," in Springer, 1996 (info) Real Analysis and Probability Richard M. Dudley, "Real Analysis and Probability," in Cambridge University Press, 2002 (info) Fundamentals of Convex Analysis Jean-Baptiste and Hiriart-Urrut and Claude Lemarechal, "Fundamentals of Convex Analysis," in Springer, 2001 (buy)

Study of Truncated Toeplitz Operators (05/07/2009 12:28 PM)

Truncated Toeplitz Operators arise in various applications including optimal input design for System Identification. Zero Displacement Rank Extension of Truncated Toeplitz Operators: Download notes pdf

On the computation of a Gramian and its inverse. (06/03/2009 05:57 PM)

Inner products can be defined in terms of how they act on basis vectors. This is usually done by writing the results of the inner product of the j-th with the i-th basis vector as the (ij)-element of a matrix. This matrix is what we refer to as Gramian. In function spaces inverting this Gramian is analogous to computing the kernel reproducing the space with respect to this inner product. Using a state-space polynomial approach we derive the classical Christoffel-Darboux formula and comment on its relations to Hankel and Bézout matrices. On the Aitken-Berg Collar Lemma: Download notes pdf, Relating the Reproducing Kernel to a two variable polynomial representing an intertwining map: Download slides pdf Comments on Matrix Representations of Residual Form Theory: Download slides pdf Worked out example in Continuous Time: Download notes pdf

Various Applications of Hankel operators (12/11/2007 08:42 PM)

Hankel operators are due to the German mathematician Hermann Hankel (1839-1873). In system theory they are often related to the idea of splitting time into past and future. So it is not very surpsing that they can be used to construct state space realizations of transfer functions. Another important application is reduced order modelling (keywords are Nevanlinna-Pick and Hankel-norm approximation). One may regard Hankel operators as module-homomorphisms which leads to many interesting insights into their internal structure as well as computational algorithms. The corresponding ring of scalars varies within applications -- but the general ideas don't. This is why we present several applications to convince the reader. In the algebraic setup we will treat a general field F inducing polynomials F[z] and so forth: Introduction to Common Notations: Download slides pdf, Regarding the Discrete Time SISO Realization Problem: Download paper pdf In the analytic setup we will focus on continuous times as well as discrete time cases in the frequency domain -- i.e., we're working with the corresponding Hilbert space L2 as well Hardy space H2 and their Banach spaces of operators: Download paper (very incomplete): pdf

Polynomial Models, Rational Models and Shifts (12/10/2007 10:51 PM)

The following slideS are not ment to be very pedagogical. I just wanted to analyze in what way the structures defined in Fuhrmann's book "A Polynomial Approach To Linear Algebra" are indeed very natural and not artificially constructed. Download slides: pdf.

Conditional Expectation (12/05/2007 02:08 PM)

Sometimes it is good to have an intuition behind abstract objects like conditional expectations. After all in daily practice we want to be able to think of conditioning as something else than the Radon-Nikodym derivative of a finite signed measure *smiles*. The following slides try to shed some intuition into the subject. Download slides: pdf.