Nevanlinna-Pick Interpolation Research Articles

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).

On rank variation of block matrices generated by Nevanlinna matrix functions

AbstractWithin the framework of the multiple Nevanlinna–Pick matrix interpolation and its related matrix moment problem, we study the rank of block moment matrices of various kinds, generalized block Pick matrices and generalized block Loewner matrices, as well as their Potapov modifications, generated by Nevanlinna matrix functions, and derive statements either on rank (or inertia) invariance in different senses or on rank variation of such types of block matrices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Sensitivity Minimization by Strongly Stabilizing Controllers for a Class of Unstable Time-Delay Systems

Weighted sensitivity minimization is studied within the framework of strongly stabilizing (stable) H-infinity controller design for a class of infinite dimensional systems. This problem has been solved by Ganesh and Pearson, [11], for finite dimensional plants using Nevanlinna-Pick interpolation. We extend their technique to a class of unstable time delay systems. Moreover, we illustrate suboptimal solutions, and their robust implementation.

Boundary Nevanlinna–Pick interpolation for Nevanlinna matrix functions and the related Hamburger matrix moment problem

This paper is concerned with a boundary Nevanlinna–Pick interpolation for Nevanlinna matrix functions and the Hamburger matrix moment problem but with specified constraints that the nonnegative matrix-valued measure has no mass distributions at a finite number of real points. Intrinsic connections between these two problems are established by the so-called Hankel vector approach. The connections, together with the theory of matrix moments and the theory of matrix polynomials with respect to a positive Hermitian block Hankel matrix due to H. Dym, enable us to get a solvability criterion and a parameterized description of all the solutions for each of these two problems.

A constrained Nevanlinna-Pick interpolation problem

We obtain necessary and sufficient conditions for Nevanlinna-Pick interpolation on the unit disk with the additional restriction that all analytic interpolating functions satisfy f'(0) = 0. Alternatively, these results can be interpreted as interpolation results for H ∞ (V), where V is the intersection of the bidisk with an algebraic variety. We use an analysis of C * -envelopes to show that these same conditions do not suffice for matrix interpolation.

Nevanlinna–Pick meromorphic interpolation: The degenerate case and minimal norm solutions

The Nevanlinna–Pick interpolation problem is studied in the class S κ of meromorphic functions f with κ poles inside the unit disk D and with ‖ f ‖ L ∞ ( T ) ⩽ 1 . In the indeterminate case, the parametrization of all solutions is given in terms of a family of linear fractional transformations with disjoint ranges. A necessary and sufficient condition for the problem being determinate is given in terms of the Pick matrix of the problem. The result is then applied to obtain necessary and sufficient conditions for the existence of a meromorphic function with a given pole multiplicity which satisfies Nevanlinna–Pick interpolation conditions and has the minimal possible L ∞ -norm on the unit circle T .

Nevanlinna-Pick Interpolation for $${\mathbb{C}} + BH^\infty$$

We study the Nevanlinna-Pick problem for a class of subalgebras of H∞. This class includes algebras of analytic functions on embedded disks, the algebras of finite codimension in H∞ and the algebra of bounded analytic functions on a multiply connected domain. Our approach uses a distance formula that generalizes Sarason’s [23] work. We also investigate the difference between scalar-valued and matrix-valued interpolation through the use of C*-envelopes.

An operator approach to multipoint Padé approximations

First, an abstract scheme of constructing biorthogonal rational systems related to some interpolation problems is proposed. We also present a modification of the famous step-by-step process of solving the Nevanlinna–Pick problems for Nevanlinna functions. The process in question gives rise to three-term recurrence relations with coefficients depending on the spectral parameter. These relations can be rewritten in the matrix form by means of two Jacobi matrices. As a result, a convergence theorem for multipoint Padé approximants to Nevanlinna functions is proved.

Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffer-type linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman.

Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler's decomposition for functions from the Schur–Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data Nevanlinna–Pick interpolation problem in the Schur class on the unit polydisk.

Interpolation problems on the spectrum of H ∞

We characterize those sequences (xn) in the spectrum of H∞ whose Nevanlinna–Pick interpolation problems admit thin Blaschke products as solutions. We also study under which conditions there is a Blaschke product B with prescribed zero-set distribution and solving problems of the form B(x) = fn(x) for every x ∈ P(xn), where P(xn) is the Gleason part associated with the point xn and where (fn) is an arbitrary sequence of functions in the unit ball of H∞. As a corollary we get a new characterization of Carleson–Newman Blaschke products in terms of bounded universal functions, a result first proved by Gallardo and Gorkin.

A generalized Stieltjes criterion for the complete indeterminacy of interpolation problems

The main result of this paper is a generalized Stieltjes criterion for the complete indeterminacy of interpolation problems in the Stieltjes class. This criterion is a generalization to limit interpolation problems of the classical Stieltjes criterion for the indeterminacy of moment problems. It is stated in terms of the Stieltjes parameters Mj and Nj. We obtain explicit formulas for the Stieltjes parameters. General constructions are illustrated by examples of the Stieltjes moment problem and the Nevanlinna-Pick problem in the Stieltjes class.

Free holomorphic functions and interpolation

In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna–Pick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function $$f : \Lambda \to \mathbb {C}$$ , where $$\Lambda$$ is an arbitrary subset of the open unit ball $$\mathbb{B}_n:=\{z\in \mathbb {C}^n: \|z\| < 1\}$$ , we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball $$[B({\mathcal H})^n]_1$$ such that $${\rm Re} \ g \geq 0 \quad {\rm and} \quad g(z)=f(z),\quad z\in \Lambda,$$ where $$B({\mathcal H})$$ is the algebra of all bounded linear operators on a Hilbert space $${\mathcal H}$$ . The proof employs several results from noncommutative multivariable operator theory and a noncommutative Cayley transform (introduced and studied in the present paper) acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions. All the results of this paper are obtained in the more general setting of free holomorphic functions with operator-valued coefficients. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball $${\mathbb B}_n$$ .

Solution of a multiple Nevanlinna–Pick problem for Schur functions via orthogonal rational functions

An interpolation problem of Nevanlinna–Pick type for complex-valued Schur functions in the open unit disk is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points. Primarily, we study the case that there exist many Schur functions fulfilling the required conditions. For this situation, an application of the theory of orthogonal rational functions is used to characterize the set of all solutions of the problem in question. Moreover, we treat briefly the case of exactly one solution and present an explicit description of the unique solution in that case.

Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna–Pick type for matrix-valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non-degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix-valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix-valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Retracts in Polydisk and Analytic Varieties with the H ∞-Extension Property

This article mainly concerns retracts in polydisk, analytic varieties with the H∞-extension property and the three-point Pick problem on \(\mathbb{D}^{3}\) . Arising in the study of Nevanlinna-Pick interpolation on the bidisk, Agler and McCarthy recently discovered a remarkable theorem which characterizes subsets in the bidisk with the polynomial extension property, and in this case, these subsets are retracts. To study H∞-extensions of holomorphic functions from subvarieties of polydisk, one naturally is concerned with retracts in polydisk. Under certain mild assumptions, it is shown that subvarieties with H∞-extension property are exactly retracts. Furthermore, we apply our argument to determine those retracts whose retractions are unique. In particular, a retract in \(\mathbb{D}^{2}\) having at least two different retractions is exactly a balanced disk. As an application, we give a sufficient condition of the uniqueness of the solution for the three-point Pick problem on \(\mathbb{D}^{3}\) .

Complete indeterminacy of the Nevanlinna-Pick problem in the class S[a,b

The class of analytic functions S[a, b] was introduced by M. G. Krein in connection with the study of a moment problem on a compact interval ([1], p. 527). The truncated Nevanlinna–Pick problem in the matrix class S[a, b] is considered in [2, 3]. In this paper we study the Nevanlinna–Pick problem in the class S[a, b] with an infinite number of complex interpolation nodes, assuming that all truncated problems are completely indeterminate. We prove that a problem with an infinite number of interpolation nodes either becomes degenerate, or remains completely indeterminate. We consider the multiplicative structure of resolvent matrices of truncated interpolation problems. We introduce generalized Stieltjes parameters. In terms of the convergence of series of Stieltjes parameters we obtain a generalized Stieltjes criterion for the complete indeterminacy of the Nevanlinna–Pick interpolation problem. We propose a special normalization of resolvent matrices of a sequence of truncated interpolation problems which guarantees their convergence to a resolvent matrix of an infinite interpolation problem. In a completely indeterminate case the set of all solutions of the Nevanlinna–Pick problem with an infinite number of interpolation nodes is described by linear-fractional transform (31).

Rational matrix functions associated with the Nevanlinna-Pick problem in the class S[a, b] and orthogonal on a compact interval

We consider the interpolation Nevanlinna-Pick problem with infinitely many interpolation nodes in the class S[a, b] and rational matrix functions associated with this problem and orthogonal on the segment [a, b]. We obtain a criterion of complete indeterminacy of the Nevanlinna-Pick problem in terms of orthogonal rational matrix functions.

Passivity-preserving model reduction by analytic interpolation

Antoulas and Sorensen have recently proposed a passivity-preserving model-reduction method of linear systems based on Krylov projections. The idea is to approximate a positive-real rational transfer function with one of lower degree. The method is based on an observation by Antoulas (in the single-input/single-output case) that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros, then the approximant is also positive real. However, this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by Byrnes, Georgiou and Lindquist, namely the maximum-entropy (central) solution. By tuning the interpolation points and the spectral zeros, as prescribed by this theory, one is able to obtain considerably better reduced-order models. We also show that, in the multi-input/multi-output case, Sorensen’s algorithm actually amounts to tangential Nevanlinna–Pick interpolation.

On the Takagi interpolation problem

Issues related to the existence and characterization of solutions of a generalization of the tangential Takagi interpolation problem are examined from the behavioral point of view. We associate to the interpolation data a “dualized” set of vector-exponential trajectories, which we model with the most powerful unfalsified model (MPUM). We show that there exists a representation of the MPUM for this set of trajectories which exhibits a special structure, and we study several properties of this representation, in particular those related to the signature of the Pick matrix of the data.