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Klaus Schiefermayr

Publications and source records attributed to Klaus Schiefermayr.

16 recordsLinked to original sources
arXiv2026

The Complex Green's Function for Symmetric Sets

The aim of this paper is threefold: (i) We study the complex Green's function (the analytic extension of the real Green's function) for multiply connected domains with some symmetry and transform it to a simple form with the help of Walsh's conformal map onto lemniscatic domains. (ii) For the complement of the union of two real intervals, we represent the complex Green's function with the help of Jacobi's elliptic and theta functions. (iii) Using this representation, we explicitly obtain all parameters of the lemniscatic domain corresponding to the complement of the two intervals. In addition, using an equality between the corresponding complex Green's functions, we obtain a numerical method for computing the conformal map from the complement of two intervals onto a lemniscatic domain which yields more accurate results than a previous method from the literature.

math.CV↗
arXiv2025

Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane

We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's criterion, the alternation theorem, and a characterization due to Rivlin and Shapiro. We derive invariance of the Widom factors of weighted Chebyshev polynomials under polynomial pre-images and a comparison result for the norms of Chebyshev polynomials corresponding to different weights. Finally, we obtain a lower bound for the Widom factors in terms of the Szegő integral of the weight function and discuss its sharpness.

math.CV↗
arXiv2024

Walsh's Conformal Map onto Lemniscatic Domains for Several Intervals

We consider Walsh's conformal map from the complement of a compact set $E = \cup_{j=1}^\ell E_j$ with $\ell$ components onto a lemniscatic domain $\widehat{\mathbb{C}} \setminus L$, where $L$ has the form $L = \{ w \in \mathbb{C} : \prod_{j=1}^\ell \lvert w - a_j \rvert^{m_j} \leq \operatorname{cap}(E) \}$. We prove that the exponents $m_j$ appearing in $L$ satisfy $m_j = μ_E(E_j)$, where $μ_E$ is the equilibrium measure of $E$. When $E$ is the union of $\ell$ real intervals, we derive a fast algorithm for computing the centers $a_1, \ldots, a_\ell$. For $\ell = 2$, the formulas for $m_1, m_2$ and $a_1, a_2$ are explicit. Moreover, we obtain the conformal map numerically. Our approach relies on the real and complex Green's functions of $\widehat{\mathbb{C}} \setminus E$ and $\widehat{\mathbb{C}} \setminus L$.

math.CV↗
arXiv2023

Walsh's Conformal Map onto Lemniscatic Domains for Polynomial Pre-images II

We consider Walsh's conformal map from the exterior of a set $E=\bigcup_{j=1}^\ell E_j$ consisting of $\ell$ compact disjoint components onto a lemniscatic domain. In particular, we are interested in the case when $E$ is a polynomial preimage of $[-1,1]$, i.e., when $E=P^{-1}([-1,1])$, where $P$ is an algebraic polynomial of degree $n$. Of special interest are the exponents and the centers of the lemniscatic domain. In the first part of this series of papers, a very simple formula for the exponents has been derived. In this paper, based on general results of the first part, we give an iterative method for computing the centers when $E$ is the union of $\ell$ intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if $E$ consists of $\ell=2$ or $\ell=3$ components satisfying certain symmetry relations then the centers and the corresponding Walsh map are given by explicit formulas. All our theorems are illustrated with analytical or numerical examples.

math.CV↗
arXiv2022

Walsh's conformal map onto lemniscatic domains for polynomial pre-images I

We consider Walsh's conformal map from the exterior of a compact set $E \subseteq \mathbb{C}$ onto a lemniscatic domain. If $E$ is simply connected, the lemniscatic domain is the exterior of a circle, while if $E$ has several components, the lemniscatic domain is the exterior of a generalized lemniscate and is determined by the logarithmic capacity of $E$ and by the exponents and centers of the generalized lemniscate. For general $E$, we characterize the exponents in terms of the Green's function of $E^c$. Under additional symmetry conditions on $E$, we also locate the centers of the lemniscatic domain. For polynomial pre-images $E = P^{-1}(Ω)$ of a simply-connected infinite compact set $Ω$, we explicitly determine the exponents in the lemniscatic domain and derive a set of equations to determine the centers of the lemniscatic domain. Finally, we present several examples where we explicitly obtain the exponents and centers of the lemniscatic domain, as well as the conformal map.

math.CV↗
arXiv2013

The Polya-Chebotarev problem and inverse polynomial images

Consider the problem, usually called the Pólya-Chebotarev problem, of finding a continuum in the complex plane including some given points such that the logarithmic capacity of this continuum is minimal. We prove that each connected inverse image $\T_n^{-1}([-1,1])$ of a polynomial $\T_n$ is always the solution of a certain Pólya-Chebotarev problem. By solving a nonlinear system of equations for the zeros of $\T_n^2-1$, we are able to construct polynomials $\T_n$ with a connected inverse image.

math.CV↗
arXiv2013

Inverse polynomial images consisting of an interval and an arc

In this paper, some geometric properties of inverse polynomial images which consist of a real interval and an arc symmetric with respect to the real line are obtained. The proofs are based on properties of Jacobi's elliptic and theta functions.

math.CV↗
arXiv2013

A lower bound for the minimum deviation of the Chebyshev polynomial on a compact real set

In this paper, we give a sharp lower bound for the minimum deviation of the Chebyshev polynomial on a compact subset of the real line in terms of the corresponding logarithmic capacity. Especially if the set is the union of several real intervals, together with a lower bound for the logarithmic capacity derived recently by A.Yu.\,Solynin, one has a lower bound for the minimum deviation in terms of elementary functions of the endpoints of the intervals. In addition, analogous results for compact subsets of the unit circle are given.

math.CV↗
arXiv2013

An upper bound for the logarithmic capacity of two intervals

The logarithmic capacity (also called Chebyshev constant or transfinite diameter) of two real intervals $[-1,α]\cup[β,1]$ has been given explicitly with the help of Jacobi's elliptic and theta functions already by Achieser in 1930. By proving several inequalities for these elliptic and theta functions, an upper bound for the logarithmic capacity in terms of elementary functions of $α$ and $β$ is derived.

math.CV↗
arXiv2013

Estimates for the asymptotic convergence factor of two intervals

Let $E$ be the union of two real intervals not containing zero. Then $L_n^r(E)$ denotes the supremum norm of that polynomial $P_n$ of degree less than or equal to $n$, which is minimal with respect to the supremum norm provided that $P_n(0)=1$. It is well known that the limit $κ(E):=\lim_{n\to\infty}\sqrt[n]{L_n^r(E)}$ exists, where $κ(E)$ is called the asymptotic convergence factor, since it plays a crucial role for certain iterative methods solving large-scale matrix problems. The factor $κ(E)$ can be expressed with the help of Jacobi's elliptic and theta functions, where this representation is very involved. In this paper, we give precise upper and lower bounds for $κ(E)$ in terms of elementary functions of the endpoints of $E$.

math.CV↗
arXiv2013

A lower bound for the norm of the minimal residual polynomial

Let $S$ be a compact infinite set in the complex plane with $0\notin{S}$, and let $R_n$ be the minimal residual polynomial on $S$, i.e., the minimal polynomial of degree at most $n$ on $S$ with respect to the supremum norm provided that $R_n(0)=1$. For the norm $L_n(S)$ of the minimal residual polynomial, the limit $κ(S):=\lim_{n\to\infty}\sqrt[n]{L_n(S)}$ exists. In addition to the well-known and widely referenced inequality $L_n(S)\geqκ(S)^n$, we derive the sharper inequality $L_n(S)\geq2κ(S)^n/(1+κ(S)^{2n})$ in the case that $S$ is the union of a finite number of real intervals. As a consequence, we obtain a slight refinement of the Bernstein--Walsh Lemma.

math.CV↗
arXiv2013

Geometric properties of inverse polynomial images

Given a polynomial $\T_n$ of degree $n$, consider the inverse image of $\R$ and $[-1,1]$, denoted by $\T_n^{-1}(\R)$ and $\T_n^{-1}([-1,1])$, respectively. It is well known that $\T_n^{-1}(\R)$ consists of $n$ analytic Jordan arcs moving from $\infty$ to $\infty$. In this paper, we give a necessary and sufficient condition such that (1) $\T_n^{-1}([-1,1])$ consists of $ν$ analytic Jordan arcs and (2) $\T_n^{-1}([-1,1])$ is connected, respectively.

math.CV↗
arXiv2013

Inequalities for the Jacobian elliptic functions with complex modulus

Despite the fact that there is a huge amount on papers and books devoted to the theory of Jacobian elliptic functions, very little is known when the modulus $k$ of these functions lies outside the unit interval $[0,1]$. In this note, we prove some simple inequalities for the absolute value of Jacobian elliptic functions with complex modulus.

math.CV↗
arXiv2013

A density result concerning inverse polynomial images

In this paper, we consider polynomials of degree $n$, for which the inverse image of $[-1,1]$ consists of two Jordan arcs. We prove that the four endpoints of these arcs form an ${\cal O}(1/n)$-net in the complex plane.

math.CV↗