arXiv subjects

Izchak Lewkowicz

Publications and source records attributed to Izchak Lewkowicz.

At least 19 recordsLinked to original sources

arXiv2026

Quantitatively hyper-positive real rational functions IV: The canonical case

In the linear time-invariant framework, passive systems are modeled by positive real functions. Dissipative systems can be modeled by the subset of quantita- tively Hyper-positive real functions, related through nested inclusions. This family was introduced and studied in our three previous works. Here, we further focus our attention on the proper subset of canonical Hyper-Positive functions. Although this family is "small", its exploration is well motivated: First, this set turns to be associated with absolute stability (the Lurie problem). Then, a systematic parametrization of all canonical Hyper-Positive functions is introduced. Moreover each canonical Hyper-Positive function can be viewed as an extreme point of the convex set of Hyper-Positive functions. Specifically a convex combination of canonical Hyper-Positive is Hyper-Positive, but not canonical. These observations hold in both frameworks: of analytic functions and of state-space realization arrays. Technically, some of the analysis is facilitated by employing Quadratic Matrix Inclusions of both, matrices and of matrix-valued rational functions.

math.CV↗

arXiv2026

Hyperpositive functions, sector bounded functions and a trace formula

We study an indexed family of functions closely related to functions ana- lytic and with a real positive part in the right open half plane (the so-called positive functions) and to the functions analytic in the right open half-plane and bounded in modulus by one there (the so-called bounded functions). These two families are related by the Cayley transform. In the present paper we introduce an affine linear relation- ship between a subclass of positive functions and the family of bounded functions, and study the corresponding connections with passivity of linear systems, interpolation and operator models. This is therefore a multidisciplinary paper, with potential readers from engineering, linear system theory and operator theory, and some repetitions of known results are given to allow various audiences to read the work. Reproducing kernel Hilbert spaces of analytic functions are a key tool in the arguments. A special role is played by the de Branges-Rovnyak spaces associated to bounded functions, and we prove a related trace formula connecting an underlying pair of operators.

math.FA↗

arXiv2026

Operator model and a trace formula for pairs of unitary operators

Using the theory of reproducing kernel Hilbert spaces introduced by L. de Branges and J. Rovnyak we prove a trace formula for pairs of operators in Hilbert space in terms of a Carathéodory function. We consider the special case, when the latter is rational. An application to the theory of first order discrete systems is given.

math.FA↗

arXiv2026

Quantitatively hyper-positive real rational functions III

Hyper-Positive Real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of Hyper-positive functions, related through nested inclusions, are introduced. Structurally, this family of functions turns out to be matrix-convex and closed under inversion. A state-space characterization of these functions through a corresponding Kalman-Yakubovich-Popov Lemma, is given. Technically, the classical Linear Matrix Inclusions, associated with passive systems, are here substituted by Quadratic Matrix Inclusions.

math.OC↗

arXiv2022

Representation theory and multilevel filters

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ setting. This is done in a framework of {\em iterated function system} (IFS) measures; these include all cases studied so far, and in particular the Julia set/measure cases. Every IFS has a fixed order, say $N$, and we show that the wavelet filters are indexed by the infinite dimensional group $G$ of functions from $X$ into the unitary group $U_N$. We call $G$ the loop group because of the special case of the unit circle.

math.FA↗

arXiv2022

Discrete Wiener Algebra in the Bicomplex Setting, Spectral Factorization with Symmetry, and Superoscillations

In this paper we present parallel theories on constructing Wiener algebras in the bicomplex setting. With the appropriate symmetry condition, the bicomplex matrix valued case can be seen as a complex valued case and, in this matrix valued case, we make the necessary connection between classical bicomplex analysis and complex analysis with symmetry. We also write an application to superoscillations in this case.

math.CV↗

arXiv2021

On the hyper-lyapunov inclusions

Gantmacher-Lyapunov Theorem (1950's) characterizes matrices whose spectrum lies in the right-half of the complex plane. Here this result is refined to Hyper-Lyapunov inclusion for matrices whose spectrum lies in some disks within the right-half plane. These disks turn to be closed under inversion, and when their radius approaches infinity, the original result is recovered. Hyper-Lyapunov inclusions are formulated through Quadratic Matrix Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within a sub-unit disk. As a by-product, it is shown that these disks closed under inversion, are a natural tool to understanding the Matrix Sign Function iteration scheme, used in matrix computations.

math.FA↗

arXiv2021

Passive Linear Discrete-Time Systems: Characterization through Structure

We here show that the family of finite-dimensional, discrete-time, passive, linear time-invariant systems can be characterized through the structure of maximal, matrix-convex set, closed under multiplication among its elements. Moreover, this observation unifies three setups: (i) difference inclusions, (ii) matrix-valued rational functions, (iii) realization arrays associated with rational functions. It turns out that in the continuous-time case, the corresponding structure is of a maximal matrix-convex, cone, closed under inversion.

math.OC↗

arXiv2020

Passive Linear Continuous-Time Systems: Characterization Through Structure

We here show that the family of continuous-time linear systems (of prescribed dimensions) can be characterized through the structure of maximal, matrix-convex, cones, closed under inversion. Moreover, this observation unifies three setups: (i) differential inclusions, (ii) matrix-valued rational functions, (iii) realization arrays associated with rational functions. It turns out that in the discrete-time case, the corresponding structure is of a maximal matrix-convex set, closed under multiplication among its elements

math.OC↗

arXiv2020

On Pseudo-Spectral Factorization over the Complex Numbers and Quaternions

This paper is a continuation of the research of our previous work and considers quaternionic generalized Carathéodory functions and the related family of generalized positive functions. It is addressed to a wide audience which includes researchers in complex and hypercomplex analysis, in the theory of linear systems, but also electric engineers. For this reason it includes some results on generalized Carathéodory functions and their factorization in the classic complex case which might be of independent interest. An important new result is a pseudo-spectral factorization and we also discuss some interpolation problems in the class of quaternionic generalized positive functions.

math.CV↗

arXiv2019

Quantitatively Hyper-Positive Real Functions

Hyper-Positive real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of Hyper-positive functions, related through nested inclusions, are introduced. Structurally, this family of functions turns out to be matrix-convex and closed under inversion. A state-space characterization of these functions, through a corresponding Kalman-Yakubovich-Popov Lemma is given. Technically, the classical Linear Matrix Inclusions, associated with passive systems, are here substituted by Quadratic Matrix Inclusions.

math.OC↗

arXiv2018

Realization of tensor-product and of tensor-Factorization of rational functions

We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of "inflating" the dimensions, we recover the classical expressions for realization of a regular product of rational functions. Then, under an additional assumption that the limit at infinity of a given rational function exists and is equal to identity, an explicit formula for a tensor-factorization of this function, is introduced.

math.OC↗

arXiv2018

Composition of rational functions: state-space realization and applications

We define two versions of compositions of matrix-valued rational functions of appropriate sizes and whenever analytic at infinity, offer a set of formulas for the corresponding state-space realization, in terms of the realizations of the original functions. Focusing on positive real functions, the first composition is applied to electrical circuits theory along with introducing a connection to networks of feedback loops. The second composition is applied to Stieltjes functions.

math.CV↗

arXiv2017

"Wrong" Side Interpolation by Low Degree Positive Real rational Functions

Using polynomial interpolation, along with structural properties of the family of positive real rational functions, we here show that a set of m nodes in the open left half of the complex plane, can always be mapped to anywhere in the complex plane by rational positive real functions whose degree is at most m. Moreover, we introduce an easy-to-find parametrization in $R^{2m+3}$ of a large subset of these interpolating functions.

math.OC↗

arXiv2016

W-Markov measures, transfer operators, wavelets and multiresolutions

In a general setting we solve the following inverse problem: Given a positive operators $R$, acting on measurable functions on a fixed measure space $(X,\mathcal B_X)$, we construct an associated Markov chain. Specifically, starting with a choice of $R$ (the transfer operator), and a probability measure $μ_0$ on $(X, \mathcal B_X)$, we then build an associated Markov chain $T_0, T_1, T_2,\ldots$, with these random variables (r.v) realized in a suitable probability space $(Ω,\mathcal F, \mathbb P)$, and each r.v. taking values in $X$, and with $T_0$ having the probability $μ_0$ as law. We further show how spectral data for $R$, e.g., the presence of $R$-harmonic functions, propagate to the Markov chain. Conversely, in a general setting, we show that every Markov chain is determined by its transfer operator. In a range of examples we put this correspondence into practical terms: $(i)$ iterated function systems (IFS), $(ii)$ wavelet multiresolution constructions, and $(iii)$ IFSs with random control.

math.PR↗

arXiv2015

A new realization of rational functions, with applications to linear combination interpolation

We introduce the following linear combination interpolation problem (LCI): Given $N$ distinct numbers $w_1,\ldots w_N$ and $N+1$ complex numbers $a_1,\ldots, a_N$ and $c$, find all functions $f(z)$ analytic in a simply connected set (depending on $f$) containing the points $w_1,\ldots,w_N$ such that \[ \sum_{u=1}^Na_uf(w_u)=c. \] To this end we prove a representation theorem for such functions $f$ in terms of an associated polynomial $p(z)$. We first introduce the following two operations, $(i)$ substitution of $p$, and $(ii)$ multiplication by monomials $z^j, 0\le j < N$. Then let $M$ be the module generated by these two operations, acting on functions analytic near $0$. We prove that every function $f$, analytic in a neighborhood of the roots of $p$, is in $M$. In fact, this representation of $f$ is unique. To solve the above interpolation problem, we employ an adapted systems theoretic realization, as well as an associated representation of the Cuntz relations (from multi-variable operator theory.) We study these operations in reproducing kernel Hilbert space): We give necessary and sufficient condition for existence of realizations of these representation of the Cuntz relations by operators in certain reproducing kernel Hilbert spaces, and offer infinite product factorizations of the corresponding kernels.

math.FA↗

arXiv2015

Realizations of infinite products, Ruelle operators and wavelet filters

Using the notions and tools from realization in the sense of systems theory, we establish an explicit and new realization formula for families of infinite products of rational matrix-functions of a single complex variable. Our realizations of these resulting infinite products have the following four features: 1) Our infinite product realizations are functions defined in an infinite-dimensional complex domain. 2) Starting with a realization of a single rational matrix-function $M$, we show that a resulting infinite product realization obtained from $M$ takes the form of an (infinite-dimensional) Toeplitz operator with a symbol that is a reflection of the initial realization for $M$. 3) Starting with a subclass of rational matrix functions, including scalar-valued corresponding to low-pass wavelet filters, we obtain the corresponding infinite products that realize the Fourier transforms of generators of $\mathbf L_2(\mathbb R)$ wavelets. 4) We use both the realizations for $M$ and the corresponding infinite product to produce a matrix representation of the Ruelle-transfer operators used in wavelet theory. By matrix representation we refer to the slanted (and sparse) matrix which realizes the Ruelle-transfer operator under consideration.

math.CV↗

arXiv2015

Characterizations of rectangular (para)-unitary rational Functions

We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) Through the realization matrix of Schur stable systems. (ii) The Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters. (iii) Through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases.

math.CV↗