Bitangential Interpolation Research Articles

Interpolation with constraint on frequency region and systems & control application

This paper elaborates on a bitangential interpolation problem with a norm constraint of the interpolant on a given frequency region (and on its generalization), and proves that it admits a solution. A motivation from systems & control for the problem is given, and the solution is applied for construction of a stabilizing control for one-block plants such that the closed-loop transfer matrix is norm-constrained on the frequency region. Illustrative numerical examples are given.

A Generalized Schur–Takagi Interpolation Problem

A generalized bitangential interpolation problem of Nevanlinna–Pick type in the class of generalized Schur matrix valued functions (mvf’s) is considered both in the unit disc and in the right half plane. Linear fractional representations of the set of solutions to these problems are presented in the strictly completely indeterminate case and in the singular case. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in Derkach and Dym (Integral Equations Operator Theory 65:1–50, 2009). The set of solutions to the problem is parametrized by the set of Schur mvf’s, which satisfy some coprimeness conditions with respect to a given pair of inner mvf’s. Enroute, it is shown that the de Branges identity for the indefinite inner product based on a self-adjoint invertible operator Q holds if and only if Q is a solution of a Lyapunov–Stein equation. This extends an observation that was made earlier in a finite dimensional setting to Pontryagin spaces.

All Solutions of a Bitangential Interpolation Problem that Includes Boundary Points

All solutions of a matrix interpolation problem with purely imaginary interpolation points are given, by a descriptor system realization, if necessary and sufficient conditions are satisfied. The first step is to reduce the interpolation problem to a minimal realization. No assumptions are needed in the paper.

Bitangential Interpolation in Generalized Schur Classes

Bitangential interpolation problems in the class of matrix valued functions in the generalized Schur class are considered in both the open unit disc and the open right half plane, including problems in which the solutions is not assumed to be holomorphic at the interpolation points. Linear fractional representations of the set of solutions to these problems are presented for invertible and singular Hermitian Pick matrices. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in Derkach and Dym (On linear fractional transformations associated with generalized J-inner matrix functions. Integ Eq Oper Th (2009, in press) arXiv:0901.0193).

Realization and interpolation for Schur–Agler-class functions on domains with matrix polynomial defining function in [formula omitted

We consider a bitangential interpolation problem for operator-valued functions defined on a general class of domains in C n (including as particular cases, Cartan domains of types I–III) which satisfy a type of von Neumann inequality associated with the domain. We show that any such function has a realization in terms of a unitary colligation and the defining polynomial for the domain. We show how the solution of various classes of bitangential interpolation problems for this class of functions corresponds to a unitary extension of a particular partially defined isometry uniquely specified by the interpolation data. Criteria for existence of solutions are given (1) in terms of positivity of a certain kernel completely determined by the data, or, more generally, (2) by the existence of a positive-kernel solution of a certain generalized Stein equation completely determined by the data.

A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space

We solve a bitangential interpolation problem for contractive multipliers on the Arveson space with an arbitrary interpolating set in the closed unit ball \( \mathbb{B}^{d}\, of\, \mathbb{C}^{d} \). The solvability criterion is established in terms of positive kernels. The set of all solutions is parametrized by a Redheffer transform.

Criteria for the strong regularity of J-inner functions and γ-generating matrices

The class of left and right strongly regular J-inner mvf's plays an important role in bitangential interpolation problems and in bitangential direct and inverse problems for canonical systems of integral and differential equations. A new criterion for membership in this class is presented in terms of the matricial Muckenhoupt condition (A2) that was introduced for other purposes by Treil and Volberg. Analogous results are also obtained for the class of γ-generating functions that intervene in the Nehari problem. The new criterion is simpler than the criterion that we presented earlier. A determinental criterion is also presented.

On a bitangential interpolation problem for contractive-valued functions on the unit ball

We solve the bitangential interpolation problem with a finite number of interpolation nodes for a multivariable analogue of the Schur class consisting of matrix-valued analytic functions on the ball. The interpolation conditions are formulated via a generalized functional calculus with operator argument, thereby generalizing in a compact way the simple, first-order interpolation conditions considered for this class of functions in earlier work. The description of all solutions is given via a Redheffer transform whose entries are given explicitly in terms of the interpolation data.

Matricial Nehari Problems, J-Inner Matrix Functions and the Muckenhoupt Condition

The classes of regular and strongly regular γ-generating matrices and J-inner matrix valued functions arose in the investigation of the matricial Nehari problem, bitangential interpolation problems, and inverse problems for canonical systems as well as the theory of characteristic functions of operators and operator nodes. In this paper, new characterizations of these classes are developed. In particular, the property of strong regularity is characterized in terms of a matricial Muckenhoupt (A2) condition in the Treil–Volberg form. These results are based on parametrizations that are intimately connected with Darlington representations of matrix valued functions in the Schur and Carathéodory classes. As a byproduct of this analysis, examples of strongly regular γ-generating matrices and entire J-inner matrix valued functions that are unbounded on the circle and the real line, respectively, are presented.

Interpolation theory in sectorial Stieltjes classes and explicit system solutions

We introduce sectorial classes of matrix-valued Stieltjes functions in which we solve the bitangential interpolation problem of Nudelman and Ball–Gohberg–Rodman. We consider also a new type of solutions of Nevanlinna–Pick interpolation problems, so-called explicit system solutions generated by Brodskii–Livsic colligations, and find conditions on interpolation data of their existence and uniqueness. We point out the connections between sectorial Stieltjes classes and sectorial operators, and find out new properties of the classical Nevanlinna–Pick interpolation matrices (in the scalar case). We present in terms of interpolation data the exact formula for the angle of sectoriality of the main operator in the explicit system solution as well as the criterion for this operator to be extremal.The interpolation model for nonselfadjoint matrices is established.

On two-sided interpolation for upper triangular matrices

The space of upper triangular matrices with Hilbert{Schmidt norm can be viewed as a nite dimensional analogue of the Hardy space H2 of the unit disk when one introduces the adequate notion of \point evaluation. A bitangential interpolation problem in this setting is studied. The description of all solution in terms of Beurling{Lax representation is given.

On Bitangential Interpolation in the Time-Varying Setting for Hilbert–Schmidt Operators: The Continuous Time Case

The Hilbert space of lower triangular Hilbert–Schmidt operators on the real line is a natural analogue of the Hardy space of a half-plane, where the complex numbers are now replaced by matrix-valued functions. One can associate with a bounded operator its “values” at a matrix-valued function [see Ballet al.,Oper. Theory Adv. Appl.56(1992), 52–89], and this allows [see Ballet al.,Integral Equations Operator Theory20(1994), 1–43] to define and solve the analogue of the two-sided Nudelman interpolation problem for bounded operators (which form an analogue ofH∞(C+)). In this paper we consider the two-sided interpolation problem with a Hilbert–schmidt norm constraint (rather than the more common operator-norm constraint) on the interpolant.

On degenerate interpolation, entropy and extremal problems for matrix Schur functions

We consider a general bitangential interpolation problem for matrix Schur functions and focus mainly on the case when the associated Pick matrix is singular (and positive semidefinite). Descriptions of the set of all solutions are given in terms of special linear fractional transformations which are obtained using two quite different approaches. As applications of the obtained results we consider the maximum entropy and the maximum determinant extension problems suitably adapted to the degenerate situation.

On a new positive extension problem for block Toeplitz matrices

We consider positive extension problems for block Toeplitz matrices when the specified entries form a stalactite type pattern. These problems do not seem in general to be amenable to the classical methods (they correspond to bitangential interpolation problems in the class of Carathéodory functions of a kind usually not solvable by the classical methods of interpolation). We solve these problems by reduction to a tangential interpolation with symmetries in a Carathéodory class.

J-inner matrix functions, interpolation and inverse problems for canonical systems, I: Foundations

This is the first of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted largely to foundational material (much of which is of independent interest) on the theory of assorted classes of meromorphic matrix valued functions. Particular attention is paid to the structure of J-inner functions and connections with bitangential interpolation problems and reproducing kernel Hilbert spaces. Some new characterizations of regular, singular and strongly regular J-inner functions in terms of the associated reproducing kernel Hilbert spaces are presented.

Dissipative periodic systems and symmetric interpolation in Schur classes

Matrix-valued functions analytic and contractive in the open unit disk (Schur functions) play an important role in system theory. They represent transfer functions of causal time-invariant dissipative systems. In this paper we show how a Schur function can still be associated to a k-periodic system. This function satisfies a certain symmetry condition (conditions (1.7) for k=2 and (4.4) for general k). We study a general bitangential interpolation problem for the Schur functions satisfying this symmetry condition.

Nudelman interpolation and the band method

In previous work the authors developed a new addition of the band method based on a Grassmannian approach for solving a completion/extension problem in a general, abstract framework. This addition allows one to obtain a linear fractional representation of all solutions of the abstract completion problem from special extensions which are not necessarily band extensions (for the positive case) or triangular extensions (for the contractive case). In this work we extend this framework to a somewhat more general setting and show how one can obtain formulas for the required special extensions from solutions of a system of linear equations. As an application we show how the formalism can be applied to the bitangential Nevanlinna-Pick interpolation problem, a case which, up to now, was not amenable to the band method.

Bitangential interpolation problems for symmetric rational matrix functions

We develop the zero-pole structure of rational matrix functions ⊖( s) that satisfy an identity of the form ⊖(s) TV⊖(s) = V for some square matrix V for which V = αV T , where α = ±1. As an application we describe the set of symmetric or antisymmetric solutions of a set of bitangential interpolation conditions. Solutions of bitangential interpolation conditions satisfying other types of symmetries are also discussed.

Bitangential interpolation for input-output maps of time-varying systems: The continuous time case

This paper introduces and presents the solution of the time-varying analogue of the two-sided Nudelman interpolation problem. As a first step the related Lagrange-Sylvester interpolation problem is solved.