Christer Glader

Approximation of delay systems—a case study

Abstract The approximation of delay systems is studied. A numerically well-behaved method for computing Hankel optima) rational approximations for delay systems is presented based on certain properties of Hankel matrices and the theory of Laguerre polynomials. The CF method of Trefethen and certain Pade approximations of delay systems are also considered. The importance of a certain Wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the L∞ norm and the Hankel norm is shown, completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions. A case study and numerical comparison is presented for the approximation of the parametric family of first-order stable delay systems G(s) = exp ( —τs)/(Ts + 1). Numerical experience indicates that L°° optimized CF approximations based on short truncated Maclaurin series give, in general, somewhat smaller L∞ approximation errors than Hankel optim...

Finite Blaschke product interpolation on the closed unit disc

Abstract We show how to construct all finite Blaschke product solutions and the minimal scaled Blaschke product solution to the Nevanlinna–Pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data. Based on a result of Jones and Ruscheweyh we note that there always exists a finite Blaschke product of degree at most n−1 that maps n distinct points in the closed unit disc, of which at least one is on the unit circle, into n arbitrary points in the closed unit disc, provided that the points inside the unit circle form a positive semi-definite Pick matrix of full rank. Finally, we discuss a numerical limiting procedure.

Rational Unimodular Interpolation on the Unit Circle

We consider an interpolation problem with n pairwise distinct nodes z1,…,zn and n numbers w1,…,wn, all on the unit circle in the complex plane, and seek interpolants b(z) of minimal degree in the class consisting of ratios of finite Blaschke products. The focus is on cases where the interpolant of minimal degree is uniquely determined.

A new algorithm for meromorphic Nevanlinna-Pick interpolation

Summary.Let D be be the open unit disc, H∞0 the space of all bounded analytic functions in D and H∞k the set of all functions of the form f(z)/(z−z1)...(z−zk) where z1,...,zk ∈ D and f ∈ H∞0. Given {z1,...,zn},{w1,...,wn}, where zi ∈D,wi ∈ and zi≠zj if i≠j, we show for 0≤k≤n−1, under certain assumptions, how to construct the unique interpolating function Bk∈H∞k, Bk(zj)=wj, of minimal essential supremum norm on ∂D by solving an eigenvalue problem defined by the interpolation data. The function Bk is a scaled quotient of two finite Blaschke products.

A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval

A method for construction of CF approximants in some cases of rational approximation of a rational function f on the unit disk and on the unit interval is presented. The inverted square root of the greatest positive eigenvalue and a corresponding eigenvector of an eigenvalue problem defined by the coefficients of f gives the solution.

An exchange algorithm for laurent polynomial chebyshev approximation on the unit circle

An exchange algorithmi for weighted Laurent polynomial Chebyshev approximation of continuous functions on the unit circle in the complex plane is presented The algorithm is at least linearly convergent. The weighted real and imaginary parts are approximated separately by weighted trigonometric polynomials using the real Reinez algorithm and a near-best Laurent Polynomial approximation is obtained. Also weighted polynomial approximation of functions analytic in the open and continuous in the closed disc is treated and comparisons CF approximation are made.

An equloscillation characterization of finite blaschke products

It is shown that rational functions continuous on the unit circle and of maximal equios-cillation order are either nonconstant finite Blaschke products or nonconstant inverted finite Blaschke products.

Crouzeix's Conjecture Holds for Tridiagonal 3 x 3 Matrices with Elliptic Numerical Range Centered at an Eigenvalue

Crouzeix stated the following conjecture in [Integral Equations Operator Theory, 48 (2004), pp. 461-477]: For every n x n matrix A and every polynomial p, |p(A)| le 2 max_z in W(A)|p(z)|, where W(A...

Non-Linear Rational Riemann-Hilbert-Problems with Circular Target Curves

This article may be considered as a continuation of [3], where we studied non-linear Riemann-Hilbert problems with circular target curves ¦w − c¦ = r and Hölder continuous coefficients c and r. Here we assume that c and r2 are rational functions and emphasize algorithmic and numerical aspects. It is shown that all solutions of the problem are rational and can be obtained by solving an interpolation problem of (generalized) Nevanlinna-Pick type. This problem is in turn reduced to a linear system, which leads to efficient (and stable) numerical methods. Special emphasis is on the Laurent case, which is of importance in applications. We propose an a- posteriori estimate which allows one to verify the accuracy of the approximate solution and report on some test calculations.