Nevanlinna-Pick Interpolation Research Articles

Nevanlinna–Pick interpolation on distinguished varieties in the bidisk

This article treats Nevanlinna–Pick interpolation in the setting of a special class of algebraic curves called distinguished varieties. An interpolation theorem, along with additional operator theoretic results, is given using a family of reproducing kernels naturally associated to the variety. The examples of the Neil parabola and doubly connected domains are discussed.

A Modified Homotopy Method for Nevanlinna-Pick Interpolation with Degree Constraint

Nevanlinna-Pick interpolation with degree constraint can be applied to control theory, signal processing and circuit theory. This paper presents a modified homotopy method to solve nonlinear equations induced by the interpolation problem. The proposed method for computing scalar real rational Nevanlinna-Pick interpolants of a bounded degree includes three parts: homotopy function construction, variable transformation and iteration procedure. The numerical example illustrates the efficiency of the new method.

Optimal Solutions to Robust Stabilization Problem for a Class of Systems with Parametric Uncertainty

This paper tackles the robust stabilization problem for the plant family described by the transfer function set P(s, δ, δ) whose coefficients of the denominator and the numerator polynomials are affine in a real uncertain parameter vector δ satisfying the norm constraint ||δ||p ≤ δ for p ≥ 1. It is shown that there exists a fixed controller robustly stabilizing all the members of P(s, δ, δ) if and only if its every member plant is free of unstable pole-zero cancelation. The optimal robust stabilizer is characterized in terms of a three dimensional frequency dependent Hahn-Banach minimizer. Classes of P(s, δ, δ) are characterized using Nevanlinna-Pick interpolation theory. For these classes optimal robust stabilizers can be obtained in closed forms using ℋ∞-optimization technique.

Nevanlinna–Pick interpolation by rational functions with a single pole inside the unit disk

We devise an efficient algorithm that, given points z1,…,zk in the open unit disk D and a set of complex numbers {fi,0,fi,1,…,fi,ni−1} assigned to each zi, produces a rational function f with a single (multiple) pole in D, such that f is bounded on the unit circle by a predetermined positive number, and its Taylor expansion at zi has fi,0,fi,1,…,fi,ni−1 as its first ni coefficients.

Some Applications of Higher-Order Hyperbolic Derivatives

The notion of higher-order hyperbolic derivatives was recently introduced. As the given definition does not lend to simple calculations, we first give a recursive formula to calculate them, which may be useful for a possible numerical use. Next, as a first application of hyperbolic derivatives of higher order, we develop some tools for solving a particular interpolation problem involving them and which allow us to consider the constrained Nevanlinna–Pick problem. Finally, we generalize and interpret some classical results of complex analysis in this new setting: Dieudonne’s Lemma and Rogosinski’s Lemma.

Hilbert function spaces and the Nevanlinna–Pick problem on the polydisc

In [4], Agler and McCarthy (2005) used Hilbert function spaces to study the uniqueness properties of the Nevanlinna–Pick problem on the bidisc. In this work we give a geometric procedure for constructing a Nevanlinna–Pick problem on D n with a specified set of uniqueness. On the way to establishing this procedure, we prove a result about Hilbert function spaces and partially answer a question posed by Agler and McCarthy.

Algorithms for solving a class of problems of H ∞-optimization of control systems

Problems of synthesis of feedbacks providing H∞-norm-maximal suppression of disturbances, acting on a linear stationary SISO system with account of limitedness of control resources, are considered. Two methods for solving are proposed. The first one is based on the Nevanlinna-Pick interpolation, and the second one is constructed on the ground of guaranteeing singularities of controllers. Within the framework of these approaches, simple computation algorithms that do not require solving Riccati equations or linear matrix inequalities are generated. The application of the developed algorithms is illustrated by an example.

A multiple point Nevanlinna-Pick type interpolation problem with both interior and boundary data for Carathéodory matrix-valued functions

The main theme of this paper is a study of a multiple point Nevanlinna-Pick type interpolation problem with both interior and boundary data for Caratheodory matrix-valued functions. The so-called Toeplitz vector approach is applied to expose intrinsic connections between such an interpolation problem and a certain truncated trigonometric matrix moment problem with specified constraints that the nonnegative matrix-valued measure has no mass distributions at the boundary interpolation nodes. These connections, together with recent results due to Bolotnikov and Dym, enable us to get solvability criteria for both the Nevanlinna-Pick interpolation problem and the matrix moment problem under consideration. On the basis of the developed theory of matrix moments we can derive parameterized descriptions of all the solutions of these two problems in the nondegenerate case. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

If \({\mathfrak{A}}\) is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \({\mathbb{A}_1(1)}\), then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \({\mathbb{A}_1(1)}\), and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H∞ acting on Hardy space or on Bergman space.

Minimum w-entropy interpolants for matricial Carathéodory functions and maximum determinant completions of associated block Pick matrix

The so-called modified block Toeplitz vector approach is used to connect a class of particular solutions G w for w ∈ D of a nondegenerate interpolation problem of the Nevanlinna–Pick type with a class of particular solutions F w of a certain matricial Carathéodory coefficient problem in a transparent way. This will suggest a simple approach to the minimum w-entropy interpolants and the maximum determinant completions of the associated block Pick matrix within the framework of that Nevanlinna–Pick type interpolation problem by using the known assertions corresponding to F w . It turns out that G w ( w ∈ D ) is exactly or provides us with the unique solution to these two extremal problems in a manner.

Spectral Nevanlinna-Pick and Caratheodory-Fejer problems for $n\geq3$

The Nevanlinna-Pick problem and the simplest case of the Carath'eodory-Fej'er problem on the spectral ball $Om_3$ are reduced to interpolation problems on the symmetrized three-disc $G_3.$

Robust Stability of a Class of Unstable Systems under Mixed Uncertainty

For unstable plants, the priority of control goes to the stability of synthesis, which means to find a stabilizer controller. In the case where the plant is subjected to structured and unstructured uncertainties, the stability problem becomes more crucial. The problem was solved by a conservative method based on generalized Kharitonov′s theorem and Nevanlinna‐Pick′s interpolation (NPI) technique. This paper introduces a proposed straightforward numerical approach for loop shaping the unstructured additive or multiplicative maximum uncertainty magnitudes. The approach finds controllers, which are capable of stabilizing the interval system while the uncertainty box is enlarged to its maximum dimensions. To illustrate, we introduce some numerical examples.

On the variance error in system identification

AbstractWe investigate how the noise and input power spectral densities influence the system identification performance. Consider the problem of identifying a linear system using the prediction error method, where Φ(eiω) is the ratio of the input power spectral density to the noise power spectral density. It is shown that the asymptotic estimation variance of the transfer function estimate is determined entirely by the non-causal part of Φ(eiω) and its first few derivatives evaluated at the poles of the system. As a result, the set of all input power spectral densities for which the asymptotic variance error remains constant, span an infinite dimensional linear manifold, which can be characterized by the solutions of a Nevanlinna-Pick interpolation problem. Since the solutions of the degree constrained Nevanlinna-Pick interpolation problem can be freely parameterized by the spectral zeros, it is possible to freely choose the zeros of Φ(eiω) without disturbing the asymptotic variance error. This provide some interesting insights on the small sample performance of the prediction error method.

The scalar Nevanlinna–Pick interpolation problem with boundary conditions

We show that if the Nevanlinna–Pick interpolation problem is solvable by a function mapping into a compact subset of the unit disc, then the problem remains solvable with the addition of any number of boundary interpolation conditions, provided the boundary interpolation values have modulus less than unity. We give new, inductive proofs of the Nevanlinna–Pick interpolation problem with any finite number of interpolation points in the interior and on the boundary of the domain of interpolation (the right half plane or unit disc), with function values and any finite number of derivatives specified. Our solutions are analytic on the closure of the domain of interpolation. Our proofs only require a minimum of matrix theory and operator theory. We also give new, straightforward algorithms for obtaining minimal H ∞ norm solutions. Finally, some numerical examples are given.

The Jacobi matrices approach to Nevanlinna–Pick problems

A modification of the well-known step-by-step process for solving Nevanlinna–Pick problems in the class of R 0 -functions gives rise to a linear pencil H − λ J , where H and J are Hermitian tridiagonal matrices. First, we show that J is a positive operator. Then it is proved that the corresponding Nevanlinna–Pick problem has a unique solution iff the densely defined symmetric operator J − 1 2 H J − 1 2 is self-adjoint and some criteria for this operator to be self-adjoint are presented. Finally, by means of the operator technique, we obtain that multipoint diagonal Padé approximants to a unique solution φ of the Nevanlinna–Pick problem converge to φ locally uniformly in C ∖ R . The proposed scheme extends the classical Jacobi matrix approach to moment problems and Padé approximation for R 0 -functions.

Multi-Point Degenerate Interpolation Problem for Generalized Schur Functions: Description of All Solutions

We consider the Nevanlinna-Pick-Caratheodory-Fejer interpolation problem with finitely many interpolation conditions in the class Sgk of meromorphic functions f with ∥f∥l∞(t) ≤ 1 and with κ poles inside the unit disk D. The problem has infinitely many solutions if and only if κ is greater than or equal to the number of non-positive eigenvalues (counted with multiplicities) of the Pick matrix P constructed from interpolation data. For each such κ, we describe the solution set of the problem in terms of a family of linearfractional transformations with disjoint ranges. The parameters defining this family are free and independent.

Boundary Nevanlinna–Pick interpolation via reduction and augmentation

We give an elementary proof of Sarason's solvability criterion for the Nevanlinna-Pick problem with boundary interpolation nodes and boundary target values. We also give a concrete parametrization of all solutions of such a problem. The proofs are based on a reduction method due to Julia and Nevanlinna. Reduction of functions corresponds to Schur complementation of the corresponding Pick matrices.

A constrained Nevanlinna-Pick interpolation problem for matrix-valued functions

Recent results of Davidson-Paulsen-Raghupathi-Singh give necessary and sufficient conditions for the existence of a solution to the Nevanlinna-Pick interpolation problem on the unit disk with the additional restriction that the interpolant should have the value of its derivative at the origin equal to zero. This concrete mild generalization of the classical problem is prototypical of a number of other generalized Nevanlinna-Pick interpolation problems which have appeared in the literature (for example, on a finitely-connected planar domain or on the polydisk). We extend the results of Davidson-Paulsen-Raghupathi-Singh to the setting where the interpolant is allowed to be matrix-valued and elaborate further on the analogy with the theory of Nevanlinna-Pick interpolation on a finitely-connected planar domain.

High-Resolution ARMA Estimation of Mixed Spectra

In this paper, we propose a high-resolution autoregressive moving average (ARMA) modeling technique for signals which are a sum of sinusoids embedded in colored noise. The approach is based on a special ARMA model. We show that an approximation to this model can be found through the central solution of Nevanlinna-Pick interpolation. In this context, it can reach a very fine resolution with a special arrangement of filterbank poles. A very efficient iterative algorithm will then be presented to achieve such desired arrangement. We also derive theoretical expressions for the variance of interpolation values for both continuous and mixed spectra for complex poles. Computer simulations show that the approach is very powerful in joint power spectrum and frequency estimation and provides superior performance with respect to traditional techniques.

A multi-point degenerate interpolation problem for generalized Schur functions

The Nevanlinna-Pick-Caratheodory-Fejer interpolation problem with finitely many interpolation conditions is considered in the class Sκ of meromorphic functions f with κ poles inside the unit disk D and with ‖ f‖L∞(T) 1 . Necessary and sufficient conditions for the existence and for the uniqueness of a solution are given in terms of the Pick matrix P of the problem explicitly determined from interpolation data. In particular it is shown that the problem admits infinitely many solutions if and only if κ is not less than the number of nonpositive eigenvalues of P . For κ equal to the number of nonpositive eigenvalues of P , we describe the solution set of the problem. Also we present necessary and sufficient conditions for the existence of a meromorphic function with a given pole multiplicity satisfying interpolation conditions and having the minimal possible L∞ -norm on the unit circle T . Mathematics subject classification (2010): 30E05.