# Elias WegertTechnische Universität Bergakademie Freiberg · Institute for Applied Analysis

Elias Wegert

Professor

## About

131

Publications

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Introduction

Additional affiliations

October 1984 - present

## Publications

Publications (131)

In his recent work, Bengt Fornberg describes the construction of finite difference schemes (FDS) for accurate numerical computation of higher order derivatives of analytic functions. In this note we introduce the characteristic function of these schemes and explore how it encodes properties of the FDS. Visualizations of the characteristic function...

Honoring Lawrence Zalcman’s work, the cover of this volume shows the phase plot of a function that appears in the construction of a special non-normal family of meromorphic functions. Neglecting all technicalities, we summarize the basic ideas of this construction and illustrate them by phase plots.

Mathematical calendar featuring phase portraits of complex functions.
Provides an exposition of the mathematical background and a biographical sketch of mathematicians whose work is related to the function presented.
In this issue: James Gregory, Stephen Butterworth, Johann Bernoulli, Josephine Mitchell, Willem Kapteyn, Yegor Ivanovich Zolotarev,...

In this note we visualize the Barnes G-function and some related functions emerging in formulas for Toeplitz determinants, and discuss some of their properties using phase plots.

Honoring Peter Duren’s work on harmonic functions, the cover of this volume visualizes a complex harmonic polynomial which has the maximum number of zeros among all such polynomials of degree 5. In this note we explain how properties of this function are encoded in its image and sketch some background.

Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $n\leq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds for $n=5$.

In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂ ⁿ cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n .
The proof is based on the unitary similarity of A to a compressed shift operator S B generated by...

Mathematical calendar featuring phase portraits of complex functions.
Provides an exposition of the mathematical background and a biographical
sketch of a mathematician whose work is related to the function presented.

In their LAMA'2016 paper Gau, Wang and Wu conjectured that a partial isometry $A$ acting on $\mathbb{C}^n$ cannot have a circular numerical range with a non-zero center, and proved this conjecture for $n\leq 4$. We prove it for operators with $\mathrm{rank}\,A=n-1$ and any $n$. The proof is based on the unitary similarity of $A$ to a compressed shi...

Gau, Wang and Wu in their LAMA'2016 paper conjectured (and proved for $n\leq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds also for $n=5$.

Eine Folge ist eine Funktion, die jeder nichtnegativen ganzen Zahl n einen (reellen) Wert xn zuweist. Die Zahlen xn nennt man Glieder (oder auch Elemente) der Folge.

In diesem Kapitel beschäftigen wir uns mit dem Lösen von Gleichungen und Gleichungssystemen, die eine oder mehrere Unbekannte (Variable) enthalten. Aufgaben dieses Typs werden in Olympiaden gern als „Einstiegsaufgaben“ gestellt. Dabei sind meist alle Werte der Variablen zu bestimmen, die sämtliche Gleichungen erfüllen. Typischerweise wird dies durc...

Honoring Walter Hayman’s groundbreaking work on entire functions, the cover of this volume is based on his counterexample to Wiman’s conjecture. In this note we sketch some background.

Die Aufgaben in diesem Buch zählen zu den Perlen mathematischer Wettbewerbe und zeigen die Schönheit und Eleganz mathematischer Sachverhalte bereits auf dem Schulniveau. Dabei ist dieses Werk aber mehr als eine Sammlung typischer und origineller Probleme, die von einem Team erfahrener Lehrer und Hochschullehrer ausgewählt wurden: Alle Aufgaben wurd...

``Complex Beauties 2020'' is a mathematical calendar featuring phase plots of complex functions. Explanations and short biographies of mathematicians which have been involved in exploring these functions are given on the back side of every page.

Crouzeix’s conjecture asserts that, for any polynomial f and any square matrix A, the operator norm of f(A) satisfies the estimate $$\begin{aligned} \Vert f(A)\Vert \le 2\,\sup \{|f(z)|:\ z \in W(A)\}, \end{aligned}$$
(1)
where \(W(A):=\{\langle Ax,x\rangle : \Vert x\Vert =1\}\) denotes the numerical range of A. This would then also hold for all fu...

In this paper, we establish several results related to Crouzeix's conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of r...

``Complex Beauties 2020'' is a mathematical calendar featuring phase plots of complex functions. Explanations and short biographies of mathematicians which have been involved
in exploring these functions are given on the back side of every page.

Honoring Stephan Ruscheweyh’s magnificent contributions to complex analysis in general and to geometric function theory in particular, the cover of this volume shows a (modified) phase plot of a Ruscheweyh derivative of the complex tangent function. In this note we sketch some background.

According to a classical theorem of Constantin Carathéodory, any Jordan domain G admits a unique conformal mapping onto the unit disk \(\mathbb {D}\) such that three distinguished boundary points of G have prescribed images on \(\partial \mathbb {D}\). This result can be extended to general domains when the role of boundary points is taken over by...

The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to other topics. Though existence and uniqueness of solutions are established for long, we present several new aspects which have not yet been explored to their full extent. In particular, we show that the following three problems are...

Mathematical Calendar 2018 featuring phase portraits of complex functions,
with biographical sketches of mathematicians.
Visit www.mathcalendar.net or www.visual.wegert.com for more information.

The phase plot of the function depicted on the cover of this volume is doubly periodic. In this expository paper, we discuss a canonical representation of all functions with doubly periodic phase (argument) in terms of the Weierstrass σ-function. In particular, we point out that the zeros and poles of such a function in a fundamental domain can be...

An equation with a Hardy space Toeplitz operator can be solved by Wiener–Hopf factorization. However, Wiener–Hopf factorization does not work for Bergman space Toeplitz operators. The only way we see to tackle equations with a Toeplitz operator on the Bergman space is to have recourse to approximation methods. The paper is intended as a review of a...

The authors establish an analogous uniform inequality for the finite Hilbert transform and a general family of weights with the help of an elementary but interesting method. They derive the inequality given by it M. Rosenblum and it J. Rovnyak [J. Math. Anal. Appl. 48, 708-720 (1974; Zbl 0298.44006)] as a particular case.

Mathematical Calendar 2018 featuring phase portraits of complex functions,
with biographical sketches of mathematicians.
Visit www.mathcalendar.net or www.visual.wegert.com for more information.

The paper is devoted to interrelations between boundary value problems of Riemann–Hilbert type and optimization problems in spaces of bounded holomorphic functions which are motivated by optimal filter design. A numerical method of Newton type for the iterative solution of nonlinear Riemann–Hilbert problems is adapted for solving the optimization p...

A Jordan domain G whose boundary is decomposed into four arcs \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) is said to be a quadrilateral. A circle packing filling the quadrilateral is a finite collection \(\mathcal {P}\) of circles, which are contained in the closure of G, and touch each other as well as the four boundary arcs in a prescrib...

Mathematical calendar featuring phase plots of complex functions, with biographical sketches of mathematicians

The technique of domain coloring allows one to represent complex functions as images on their domain. It endows functions with an individual face and may serve as simple and efficient tool for their visual exploration. The emphasis of this paper is on phase plots, a special variant of domain coloring. Though these images utilize only the argument (...

The note explains how snowflakes (for a greeting card) can be made from phase plots of complex functions.

Mathematical calendar featuring phase plots of complex functions, with biographical sketches of mathematicians

Mathematical calendar featuring phase plots of complex functions, with biographical sketches of mathematicians

Circle packings with specified patterns of tangencies form a discrete
counterpart of analytic functions. In this paper we study univalent packings
(with a combinatorial closed disk as tangent graph) which are embedded in (or
fill) a bounded, simply connected domain. We introduce the concept of crosscuts
and investigate the rigidity of circle packin...

We propose an algorithm for the efficient numerical computation of the periodic Hilbert transform. The function to be transformed is represented in a basis of spline wavelets in Sobolev spaces. The underlying grids have a hierarchical structure which is locally refined during computation according to the behavior of the involved functions. Under ap...

A trilateral is a Jordan domain
$G$
G
whose boundary is decomposed into three Jordan arcs
$\alpha , \beta $
α
,
β
and
$\gamma $
γ
. A circle
$C$
C
which is contained in the closure of
$G$
G
and touches
$\alpha , \beta $
α
,
β
and
$\gamma $
γ
is called an incircle of the trilateral. Using Sperner’s Lemma, we prove that ev...

This calendar provides a look into the wonderful world of complex functions and the life and work of mathematicians who contributed to our understanding of this field of mathematics. It is the fourth year of the production of “Complex Beauties.” We have added two new members to our calendar team, Ueli Daepp and Pam Gorkin. As a guest author, Bengt...

Summary: The cover of this volume shows the phase portrait of a rational function. In this note, we explain how its poles and zeros are chosen in order to create the four letters $bold C, bold M, bold F, bold T$. Moreover, we prove that phase portraits of rational functions can "visually approximate" any image composed of saturated colors.

In this paper, we summarize some facts on spline wavelets, analyze the Hilbert transform of these wavelets on the real line and on the unit circle, describe an algorithm for computing the Hilbert transform on uniform grids, and report on some test calculations.

This year's calendar, "Complex Beauties," delights both the visual senses and the intellect. These colorful representations of complex functions are fascinating aesthetically as well as mathematically. The researchers' biographical sketches and the explanations of the mathematical functions with which their work is connected provides insight into v...

Summary: Circle packings are configurations of circles satisfying specified patterns of tangency and have emerged as the foundation for a fairly comprehensive theory of discrete analytic functions. Though many classical results found their counterpart in circle packing, other concepts have not yet been transferred, particularly those which require...

Summary: We prove asymptotic formulas for the first discrete moment of the Riemann zeta function on certain vertical arithmetic progressions inside the critical strip. The results give some heuristic arguments for a stochastic periodicity that we observed in the phase portrait of the zeta function.

In Section 3.6 we have seen that the process of analytic continuation may result in “multiple-valued” functions. Notwithstanding their name, these objects are not ordinary functions and must be handled with care.

This chapter is concerned with various methods for constructing analytic functions. We begin by investigating limit processes involving analytic functions in Section 5.1. In particular we shall introduce the notion of normal convergence, which plays an important role in complex analysis. This concept will be used in Section 5.2, where we prove Mont...

In this chapter we shall get introduced to the most important entities of this book: analytic functions. Though they form just a small minority within the class of all complex functions, they are of crucial importance both in theory and in applications. Moreover, they possess a number of fascinating and somewhat unexpected properties which we shall...

Graphical representations of functions belong to the most useful tools in mathematics and its applications. While graphs of (scalar) real-valued functions can be depicted easily in a plane, the graph of a complex function in one variable is a surface in four-dimensional space. Since our imagination is trained in three dimensions, most of us have di...

The main theme of this chapter is differentiation and integration of complex functions. In Section 4.1 we introduce the concept of the derivative, on which complex calculus is based. Though there is formally no difference between the definitions of differentiability for real and complex functions, the nature of these two notions is quite different....

The functions we shall be exploring in this book are complex-valued functions of a single complex variable. When we speak of complex functions, we do not necessarily mean that these functions are analytic, although emphasis will be placed on this special and extremely important class.

The focus of this chapter is on geometric aspects: complex functions will be considered as mappings between subsets of two complex planes which transplant objects from one plane to the other.

We investigate a discrete counterpart of Beurling's boundary value problem for analytic functions in the framework of circle packing. In the case of discrete analytic functions modelled on arbitrary combinatorically closed disks existence of solutions is shown under rather general assumptions. As in the nondiscrete case finitely many branch circles...

This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations - so-called phase portraits - which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and pl...

The illustrations in this calendar show visualizations of complex functions and portraits of mathematicians whose work is connected with these functions. A short explanation of the mathematics used to create each picture as well as some biographical information of the featured mathematician appears on the back of each page. Some basic knowledge of...

Das besondere Interesse beim diesjährigen Kolloquium gilt einerseits den
aktuellen und zukünftigen Lehrinhalte im Themengebiet der ungleichmäßig
übersetzenden Getriebe und wie sich diese nach der fast vollständigen
Umstellung der deutschen Studiengänge auf das zweigliedrige Bachelor und
Masterstudium darstellen. Andererseits darf man ebenso auf die...

We study best approximation of functions in the Hardy space H 2() by Blaschke forms, which are finite linear combinations of modified Blaschke products. These functions have poles outside the unit disk which are adapted according to the function to be decomposed. We prove the existence of minimizers and propose an algorithm for their construction.

We propose to visualize complex (meromorphic) functions $f$ by their phase $P_f:=f/|f|$. Color--coding the points on the unit circle converts the function $P_f$ to an image (the phase plot of $f$), which represents the function directly on its domain. We discuss how special properties of $f$ are reflected by their phase plots and indicate several a...

In der globalisierten Welt gehören Kreativität und neue Ideen zu den wenigen Möglichkeiten, einen Wettbewerbsvorteil zu erlangen. In einem Fachgebiet- wie im vorliegenden Fall der Mechanismentechnik das zumindest in Teilen als nicht mehr zeitgemäß erachtet wird und dessen spezifisches Wissen Gefahr läuft, vergessen zu werden, bieten sich denjenigen...

The illustrations of this calendar show visualizations of complex functions and portraits of mathematicians connected with these functions. Each reverse side features a short explanation of the picture and biographical information of the depicted personality.

The subject of this paper is Beurling’s celebrated extension of the Riemann mapping theorem [5]. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem (due to Beurling)
contains a number of gaps which seem inherent in Beurling’s geometric and approximative approach. We provide a complete proof
o...

Author(s) of this paper may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr. For further information see http://journals....

In the lecture we pose and study a discrete counterpart of nonlinear boundary value problems for holomorphic functions in the framework of circle packing. The classical setting of these problems originates from to Bernhard Riemann's thesis. Though Riemann emphasized geometric aspects of the problem,
it took more than one hundred years until ideas f...

The paper demonstrates the use of phase diagrams as tools for visualizing and exploring meromorphic functions. With any such function \(f:D\ \rightarrow\ \hat{\rm C}\) we associate two mappings
$$P_{f}:D\rightarrow{\rm T}\cup \lbrace {0,\infty}\rbrace,z\mapsto {f(z)\over|f(z)|},\qquad V_{f}:D\rightarrow {\rm C},z\mapsto - {f(z){\overline f^\prime(z...

Starting from the notion of the complex pseudo-hyperbolic distance and the hyperbolic difference quotient introduced by A. F. Beardon and D. Minda in [1], we define hyperbolic divided differences for unimodularly bounded holomorphic functions in the complex unit disk and investigate their mapping properties. In particular, we show that they operate...

Summary: This article may be considered as a continuation of [it C. Glader and it E. Wegert, Math. Nachr. 281, No. 9, 1221--1239 (2008; Zbl 1230.30027)], 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 $r^2$ are rational functions...

We propose a discrete counterpart of non-linear boundary value problems for holomorphic functions (Riemann-Hilbert problems) in the framework of circle packing. For packings with simple combinatorial structure and circular target curves appropriate solvability conditions are given and the set of all solutions is described. We compare the discrete a...

The procedures studied in this paper originate from a problem posed at the International Mathematical Olympiad in 1986. We present several approaches to the IMO problem and its generalizations. In this context we introduce a “signed mean value procedure” and study “relaxation procedures on graphs”. We prove that these processes are always finite, t...

The paper gives a systematic and self-contained treatment of the nonlinear Riemann–Hilbert problem with circular target curves |w – c | = r, sometimes also called the generalized modulus problem. We assume that c and r are Hölder continuous functions on the unit circle and describe the complete set of solutions w in the disk algebra H∞ ∩ C and in t...

Fanny and Freddy are little frogs. Since they got a new MAE, which is a flexible chain made from frog spawn, their favorite avocation is measuring. Playing around with this device they accidentally (almost) discover π and e.
The story tells how this happened and explains how this discovery is related to circle packing.

We prove that every unimodularly bounded measurable function on the complex unit circle admits a representation where f+ and f- extend holomorphically into the interior and the exterior of the circle, respectively, f- vanishes at infinity, and both functions are unimodularly bounded. The representation is unique if ∞ < 1.

The authors discuss some geometric and topological aspects of holomorphic curves in loop spaces, with a special emphasis on their relations to boundary value problem of Riemann-Hilbert type. The general scheme is applied to holomorphic functions with values in the space of loops immersed in a three-dimensional Riemannian manifold ($3$-fold). Then t...

{The classical Nevanlinna-Pick interpolation problem is to find functions $f$, analytic in the unit disc $Bbb D$, which are bounded by $1$ and satisfying the interpolation conditions $f(z_i)=w_i$ ($i=1,dots,n$) for given points $z_1,dots,z_n$ in the open unit disk $Bbb D$ and complex numbers $w_1,dots,w_n$. Nevanlinna and Pick provided necessary an...

Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we prove new existence and uniqueness results for solutions of nonlinear Riemann-Hilbert problems with noncompact restriction curves.

Julius Ludwig Weisbach wurde am 10. August 1806 in Mittelschmiedeberg als achtes von neun Kindern der Eheleute Christian Gottlieb Weisbach und Christina Rebekka Weisbach, geb. Stephan, geboren. Nach der Grundschule besuchte er ein Jahr lang das Lyceum in Annaberg und schloss dieses mit der 3. Klasse ab. Mit vierzehn Jah-ren begann er eine Bergmanns...

The paper is devoted to the linearized H ‐equation of Chandrasekhar and Ambarzumyan. A Stieltjes‐type transform reduces the equation to a boundary value problem for holomorphic functions in the upper half‐plane which is solved in closed form. Additional conditions ensure that the solutions Φ extend holomorphically to the lower half‐plane slit along...

In recent years R. Belch proposed an approach for investigating nonlinear Riemann-Hilbert problems with non-smooth target
manifold. His main result is a characterization of solutions to Riemann-Hilbert problems as extremal functions in certain
function classes. However, a complete analogy to corresponding results for problems with smooth target man...

Using Fourier techniques and the inner–outer factorization of holomorphic functions a complete description of the various solutions to the autocorrelation equation on a finite interval is given.
The Fourier transform of the solutions admits an representation involving Cauchy integrals and Blaschke products. Using this representation a general effic...

Nonlinear cross-correlation equations in finite and semi-infinite intervals are studied by means of Fourier transform and Cauchy integral techniques. The equations are reduced to a bilinear conjugacy problem for two analytic functions on the real axis which can be solved in explicit form.
In particular, Engibaryan's equations from nonlinear factori...

We study properties of solutions to non-linear Riemann-Hilbert problems with smooth compact regularly traceable target manifold. The inves-tigations focus on the dependence of solutions with positive winding numbers on additional parameters. While previous results investigated the behavior of the solutions inside the unit disk, we also pay attentio...

Restrictions imposed on the boundary values of holomorphic func-tions induce restrictions on their values at interior points. The paper is devoted to the following related question: Let A be a subclass of the Hardy space H 1 in the complex unit disk D and for each t ∈ ∂D let the complex plane be divided into an upper und a lower domain by some curv...

Summary: We extend some of our previous results about the existence of analytic discs attached to totally real smooth targets to certain new situations, including totally real surfaces with isolated singular points. Existence of singular points implies some new phenomena so we begin with considering algebraic germs of two-surfaces which are totally...

{Summary: Let $f:bbfDtoØmega$ be a conformal mapping of the unit disk $bbfD$ onto a starlike domain $G$ normalized by $f(0)=0$. In this note we derive the uniform estimate $$ømega_varphi(delta)leq fracpilog R+6vert log deltavert},,$$ for the modulus of continuity $ømega_varphi(delta)$ of the boundary correspondence function $varphi:=textarg,fvert_p...

The authors present an explicit formula for the number of complex points on the graph of a polynomial endomorphism of the plane (theorem 1). They also show that one can effectively compute the number of elliptic complex points of a planar polynomial endomorphism by using a finite number of algebraic and logical operations on its coefficients (theor...

A class of nonlinear singular integral equations of Cauchy type on a finite interval is transformed to an equivalent class of (discontinuous) boundary value problems for holomorphic functions in the complex unit disk. Using recent results on the solvability of explicit Riemann–Hilbert problems, we prove the existence of solutions to the integral eq...

Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we are able to prove existence and uniqueness results for solutions of nonlinear RiemannHilbert problems with non-compact Lipschitz continuous restriction curves.

Authors' summary: We deal with analytic discs attached to (with boundaries in) totally real two-dimensional surfaces. Firstly, we prove that certain non-compact smooth totally real surfaces are globally foliated by boundaries of analytic discs. Secondly, we describe a topological approach to existence results for analytic discs attached to small ge...

.An existence theorem for solutions of non-linear transmission problems is proved using certain special estimates of inverses of Toeplitz operators. A framework for construction of analytic discs attached to singular targets is described, and a mechanism of creating such discs is presented. reziume teplicis operatorebis SeuGlebulebis specialuri SeP...

Summary: We propose a one-dimensional model for the vorticity equation involving viscosity. Complex methods are utilized in order to study finite time blow-up of the solutions. In particular, it is shown that the blow-up time depends monotoneously on the viscosity.