Franck Wielonsky

A criterion for uniqueness of a critical point inH 2 rational approximation

This paper presents a criterion for uniqueness of a critical point inH2,R rational approximation of type (m, n), withm≥n-1. This criterion is differential-topological in nature, and turns out to be connected with corona equations and classical interpolation theory. We illustrate its use with three examples, namely best approximation of fixed type on small circles, a de Montessus de Ballore type theorem, and diagonal, approximation to the exponential function of large degree.

On a rational approximation problem in the real Hardy space H 2

1. Introduction The problem of finding a rational approximation to a holomorphic function in a domain 52 often arises in practice. Let us give an example from system theory which has first motivated our work [l]. System theory is concerned with the study of physical systems that can be described by a relation between inputs and outputs. When considering discrete time systems, it is a convention to denote by u(z)=Cksko ukzmk, UkER,

Riemann-Hilbert analysis and uniform convergence of rational interpolants to the exponential function

We study the asymptotic behavior of the polynomials p and q of degrees n, rational interpolants to the exponential function, defined by p(z)e-z/2 + q(z)ez/2 = O (ω2n + 1(z)), as z tends to the roots of ω2n + 1, a complex polynomial of degree 2n+1. The roots of ω2n+1 may grow to infinity with n, but their modulus should remain uniformly bounded by c log(n), c > 1/2, as n → ∞. We follow an approach similar to the one in a recent work with Arno Kuijlaars and Walter Van Assche on Hermite-Pade approximants to ez. The polynomials p and q are characterized by a Riemann-Hilbert problem for a 2 × 2 matrix valued function. The Deift-Zhou steepest descent method for Riemann-Hilbert problems is used to obtain strong uniform asymptotics for the scaled polynomials p(2nz) and q(2nz) in every domain in the complex plane. From these asymptotics, we deduce uniform convergence of general rational interpolants to the exponential function and a precise estimate on the error function. This extends previous results on rational interpolants to the exponential function known so far for real interpolation points and some cases of complex conjugate interpolation points.

A Rolle's theorem for real exponential polynomials in the complex domain

Abstract We present a version of Rolles theorem for real exponential polynomials having a number L sufficiently large of zeros in a compact set K of the complex plane. We show that the derivative of the exponential polynomials have at least L−1 zeros in a region slightly larger than K . The method of proof is elementary and similar to that of the classical Jensens theorem about the location of the zeros of the derivative of a real polynomial. The proof also relies on known results concerning the distribution of the zeros of real exponential polynomials. Besides, we display a Rolles theorem for higher-order derivatives and as a conclusion make a few comments about the maximal number of zeros a real exponential polynomials may have in a given compact set of C .

Non-uniqueness of rational best approximants

Abstract Let f be a Markov function with defining measure μ supported on (−1,1), i.e., f(z)=∫(t−z) −1 d μ(t), μ⩾0 , and supp (μ)⊆ ( −1,1) . The uniqueness of rational best approximants to the function f in the norm of the real Hardy space H 2 (V), V ≔ C ⧹ D ={z∈ C | |z|>1} , is investigated. It is shown that there exist Markov functions f with rational best approximants that are not unique for infinitely many numerator and denominator degrees n−1 and n, respectively. In the counterexamples, which have been constructed, the defining measures μ are rather rough. But there also exist Markov functions f with smooth defining measures μ such that the rational best approximants to f are not unique for odd denominator degrees up to a given one.

Weighted H 2 rational approximation and consistency

Summary. We investigate consistency properties of rational approximation of prescribed type in the weighted Hardy space

Asymptotics of Diagonal Hermite-Padé Approximants toez

H^2_-(\mu)

Asymptotic Uniqueness of Best Rational Approximants of Given Degree to Markov Functions in L 2 of the Circle

for the exterior of the unit disk, where

Asymptotic Error Estimates for L2 Best Rational Approximants to Markov Functions

\mu

Rational Approximation to the Exponential Function with Complex Conjugate Interpolation Points

is a positive symmetric measure on the unit circle