Tiberiu Constantinescu

Recursive solutions of rational interpolation problems via fast matrix factorization

We describe a novel approach to analytic rational interpolation problems of the Hermite-Fejér type, based on the fast generalized Schur algorithm for the recursive triangular factorization of structured matrices. We use the interpolation data to construct a convenient so-called generator for the factorization algorithm. The recursive algorithm then leads to a transmission-line cascade of first-order sections that makes evident the interpolation property. We also give state-space descriptions for each section and for the entire cascade.

Representations of Hermitian Kernels by Means of Krein Spaces. II. Invariant Kernels

Abstract: In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kreîn space. This is motivated by the GNS representation of *-algebras associated to hermitian functionals, the dilation theory of hermitian maps on C*-algebras, as well as others. We explain the key role played by the technique of induced Kreîn spaces and a lifting property associated to them.

Time-variant displacement structure and interpolation problems

Derives a new recursive solution for a general time-variant interpolation problem of the Hermite-Fejer type, based on a fast algorithm for the recursive triangular factorization of time-variant structured matrices. The solution follows from studying the properties of an associated cascade system and leads to a triangular array implementation of the recursive algorithm. The system can be drawn as a cascade of first-order lattice sections, where each section is composed of a rotation matrix followed by a storage element and a tapped-delay filter. Such cascades always have certain blocking properties, which can be made equivalent to the interpolation conditions. The authors also illustrate the application of the algorithm to problems in adaptive filtering, model validation, robust control, and analytic interpolation theory. >

Minimal signature in lifting of operators, II

Abstract This paper deals with a problem of lifting of operators in Krein spaces, with minimal signature numbers of the defects. There are obtained necessary and sufficient conditions for the existence of liftings and parametrization formulae for the set of all solutions. As applications there are treated a problem of selfadjoint extensions of symmetric operators and a variant of lifting of commutants.

Active optical lattice filters

Highly integrated, novel architectures for optical filtering can leverage structure, gain and variable delays to provide a multi-use photonic platform. Hardware and software results are presented.

Displacement Structure and Completion Problems

A general result is proven concerning time-variant displacement equations with positive solutions in a general operatorial setting. It is then shown that the solutions of several completion problems, recently considered in connection with classical interpolation and moment theory, follow as special cases of the main result. The main purpose of this paper is to show that under supplementary finite-dimensionality conditions, a so-called generalized Schur algorithm, which naturally arises in connection with displacement equations, can be used to prove the above-mentioned result. The associated transmission-line interpretation is also discussed in terms of a cascade of elementary sections with intrinsic blocking properties.

Parametrizing Quantum States and Channels

AbstractThis work describes one parametrization of quantum states and channels and several of its possible applications. This parametrization works in any dimension and there is an explicit algorithm which produces it. Included in the list of applications are a simple characterization of pure states, an explicit formula for one additive entropic quantity which does not require knowledge of eigenvalues, and an algorithm which finds one Kraus operator representation for a quantum operation without recourse to eigenvalue and eigenvector calculations. PACS: 03.67a, 03.67-Hk, 03.67-Lx

Composition methods for four-port couplers in photonic integrated circuitry

Planar photonic integrated circuits based on four-port couplers offer enhanced sophistication and functionality. Each four-port coupler is characterized by sixteen signal coupling coefficients governed by ten energy constraints. The ability to generate the constrained sixteen coupling coefficients is needed in the analysis of the four-port coupler. However, the energy constraint equations are nonlinear and cumbersome to solve directly. We introduce two techniques to reduce these signal coupling coefficients to a set of six free parameters. Hence we can characterize all possible couplers in terms of their sixteen constrained coupling coefficients, or either of two sets of six free parameters. This reduction in parameters has significant ramifications for the design, specification, and empirical characterization of these useful building blocks.

Inheritance principles for chordal graphs

Abstract We prove two results concerning positive completions of partial positive matrices. First, we give a parametrization of all positive invertible completions of a partial positive matrix having as support a chordal graph, along a fixed increasing chordal sequence. Then, a formula for the determinant of each completion is obtained in terms of the associated parameters. As an application, we obtained a proof of a conjecture of W. W. Barrett, C. R. Johnson, and L. Rodman regarding an inheritance principle for chordal graphs.

Invariant Hermitian kernels and their Kolmogorov decompositions

Abstract We study Hermitian kernels invariant under the action of a semigroup with involution. We characterize the Hermitian kernels that realize the given action by bounded operators on a Krein space. Applications include the GNS representation of ∗ -algebras associated to Hermitian functionals and the dilation theory of Hermitian maps on C ∗ -algebras.