Computational Methods and Function Theory
Proceedings 1997

Abstracts

A. Ambroladze: Some Open Problems in the Theory of Rational Interpolants in the Complex Plane, 001--008
We discuss some results about the convergence of rational interpolants with preassigned poles (partly or completely) to analytic functions, and some open problems related to these results. The open problems have relatively elementary formulations and, to our opinion, their solutions would give a better understanding of the subject. We discuss also some computational aspects of this theory.

G. D. Anderson, S. Qiu, M. K. Vuorinen: Bounds for the Hersch-Pfluger and Belinskii Distortion Functions, 009--022
The authors study the Hersch-Pfluger plane quasiconformal distortion function, solving extremal problems, obtaining monotoneity and convexity properties, and finding sharp bounds for it. They also obtain bounds for the Belinskii function, which measures the deviation of a quasiconformal automorphism of the unit disk from the identity map.

V. Andrievskii: Locally Convex Curves and Distribution of Zeros of Polynomials, 023--044
We introduce the class of locally Dini-convex curves which includes convex curves as well as piecewise Dini-smooth curves without cusps. We obtain estimates for the discrepancy between the equilibrium measure $\mu_L$ of a locally Dini-convex curve $L$ and the zero-counting measure $\nu_{p_n}$ of a monic polynomial $p_n$ of degree $n=1,2,\ldots$ in terms of one-sided bounds for the difference $U(\mu_L -\nu_{p_n},z)$ of their logarithmic potentials. Both situations --- when either an upper or a lower bound of $U(\mu_L -\nu_{p_n},z)$ is known --- are treated simultaneously.

L. Baratchart: Rational and Meromorphic Approximation in Lp of the Circle: System-Theoretic Motivations, Critical Points and Error Rates, 045--078
This paper is concerned with best rational or meromorphic approximation of fixed type in $L^p$ of the circle, when poles are constrained to lie in the unit disk. We first discuss some motivations from system theory and $n$-th root asymptotics for the error when the function to be approximated is analytic in a neighborhood of the circle. We then consider the problem from a differential-topological viewpoint, and derive an index theorem linking together the Morse indices of critical points. We subsequently relate Morse indices and error rates, and apply this to the issue of uniqueness of a critical point in $H^2$ rational approximation. The approach is illustrated on Markov functions and on the exponential function.

V. I. Belyi, E. V. Kravchuk: Generalized Faber Sets and Applications, 079--086
Let $B$ be a compact set in the extended complex plane~${\Bbb C}$ with a simply connected complement containing $\infty$. We consider some generalizations of the Faber transform, namely the operators $\tilde{T}: w^n \mapsto \tilde{F}_n(z)$, $n\in{\Bbb N}$, where $\tilde{F}_n(z)$ are generalized Faber polynomials for $B$ which are generated by the Hadamard composition of Faber polynomials with adjoint kernels. Here the adjoint kernels ${\cal F}(z)$ and ${\cal F}^*(z)$ are two analytic functions in the unit disk ${\Bbb D}$ with power extensions ${\cal F}(z) = \sum_{k=0}^{\infty} b_kz^k$, $b_k\neq0$, $k\in{\Bbb N}$, ${\cal F}^*(z) = \sum_{k=0}^{\infty} {b_k}^{-1} z^k$. It is supposed that ${\cal F}(z)$ and ${\cal F}^*(z)$ have analytic extensions to ${\Bbb C}$ along an arbirtrary continuous curve $\Gamma$ which does not intersect the points $1$, $\infty$, $0$. We study sufficient geometric conditions for $B$ which imply~$\Gamma$ to be bounded (that is $B$ is a so called generalized Faber set'') and consider some applications to the order estimates of various $n$th widths for some classes of analytic functions.

D. Betsakos: An Extension of the Beurling-Nevanlinna Projection Theorem, 087--090
We prove a polarization and a projection result for the harmonic measure on the unit disk. The projection result is a generalization of the Beurling-Nevanlinna projection theorem.

G. Brown, S. Koumandos: On a Theorem of S. Ruscheweyh, 091--098
A Theorem of S. Ruscheweyh states that for $\lambda \geq 1/2$, $-1 < x < 1$, and arbitrary nonincreasing sequence $a_{k}$, $k=0,1,\ldots,n$, we have
$$\sum_{k=0}^{n}a_{k}\,\frac{C_{k}^{\lambda}(x)}{C_{k}^{\lambda}(1)}z^{k}\neq 0, \qquad\left|z \right|\leq1,$$
where $C_{k}^{\lambda}$ are the Gegenbauer polynomials. We give a new proof of this result by Fourier kernel techniques. We further show that these methods enable us to obtain corresponding results for even polynomial sums of this type and specific sequences $a_{k}$. Other related results are also discussed.

D. Bshouty, W. Hengartner, M.-H. Nicole: A Constructive Method for Univalent Harmonic Exterior Maps with Blaschke Dilatation, 099--116
Necessary and sufficient conditions are established which allow the construction of univalent harmonic mappings whose second dilatation function is a given Blaschke product of degree $N$ and which maps the exterior of the unit disk onto a given domain $\Omega$ containing infinity. Explicit examples are given for the cases $N=3$ and $N=4$.

P. Caraman: New Results About the Equality Between the p-Module and the p-Capacity, 117--134
In this paper, continuing our earlier work, we establish some new cases of equality between the $p$-module and the $p$-capacity of a condenser, i.e. of a triple of sets $(E_0, E_1, D)$, where $D\subset\overline{{\Bbb R}^n}$ is a domain, $E_0, E_1\subset\bd D$, $\overline E$ is the closure of $E$ with respect to ${\Bbb R}^n$ and $\bd E$ is its closure with respect to $\overline{R^n}.$ Finally, we improve a result of this kind obtained previously by the author.

A. Cuyt, B. Verdonk: Extending the q-Algorithm to Tackle Multivariate Problems, 135--160
A lot has been said and done about the $qd$-algorithm. Our main interest is to analyze how the original algorithm and its various improvements can be generalized for use in several multivariate applications. The present paper recalls the known univariate results in the Sections 2.1 and 3, and discusses their multivariate generalization in the Sections 2.2 and 4, but without going into all the multivariate details. We just make everything multivariate-ready for implementation in floating-point polynomial arithmetic (covering additional difficulties not encountered in exact polynomial arithmetic). The reader who is only familiar with the properties of the univariate $qd$-algorithm and does not have an extensive knowledge of the multivariate theory, can easily follow the analysis.

A. A. Danielyan: M. A. Lavrentyev's Problems on Pointwise Polynomial Approximation and Related Questions, 161--170
A function $f(z)$ defined in a domain $D$ of the complex plane is called a limit function, if there exists a polynomial sequence converging to $f(z)$ everywhere in $D$. The set $E$ of singular points of the limit function is nowhere dense in $D$. In 1936 M.~A.~Lavrentyev posed a problem on the mutual connection between the values of a limit function in different domains complementary to $E$. The main purpose of the article is to give a complete solution to this problem.

M. J. Dorff: Harmonic Univalent Mappings onto Asymmetric Vertical Strips, 171--176
Let $\Omega _{\alpha}$ be the asymmetrical vertical strips defined by $\Omega _{\alpha}=\{w:\frac{\alpha -\pi} {2\sin \alpha} < \mbox{ Re }w < \frac{\alpha}{2\sin \alpha} \}$, where $\pi/2 \leq \alpha < \pi$, and let $\Bbb{D}$ be the unit disk. We characterize the class $\SH (\Bbb{D},\Omega _{\alpha})$ of univalent harmonic orientation-preserving functions $f$ which map $\Bbb{D}$ onto $\Omega _{\alpha}$ and are normalized by $f(0)=0$, $f_{\overline{z}}(0)=0$, and $f_z(0) > 0$. Then we use this characterization to demonstrate a few other results.

P. Duren: Robin Capacity, 177--190
This is a survey of recent work on Robin capacity. With respect to a given domain in the complex plane, Robin capacity measures the size of a subset of the boundary. Its potential-theoretic definition is given in terms of the Robin function of the domain, so named by Bergman and Schiffer [3] for the French mathematical physicist Gustave Robin (1855-1897). The Robin function is a generalization of Green's function and plays a similar role of resolvent kernel in the solution of mixed boundary-value problems. Like ordinary capacity, the Robin capacity can be defined in a variety of equivalent ways: through potential theory, extremal length, and least-energy considerations. In addition, it has a basic connection with conformal mapping.

H.-J. Fischer: On Generating Orthogonal Polynomials with Respect to Self-Similar Measures, 191--202
In the present paper, we derive an algorithm for computing the recurrence coefficients of orthogonal polynomials with respect to self-similar measures. This means that the cumulative distribution function of the measure is a fractal interpolation function in the sense of Barnsley. As examples show, this notion is very flexible. But here we consider only real fractals, as in the case of Cantor measure or of some special Riesz-Nagy measures.

J.-D. Fournier: Complex Zeros of Random Szegö Polynomials, 203--224
Orthogonal Szeg\H{o} polynomials are built using a measure supported by the unit circle. Here its density is taken as the energy spectrum of a {\it random\/} real signal $X(m)$, with disrete $m$, \hbox{$0\leq m < N$}. Given some probabilistic assumptions on the signal, one is after the statistics of the zeros of the associated polynomials. For \eg white gaussian signal, the probability distribution of their moduli is extremely peaked at $\Lambda = N^{-1/(2p)}$, where $p$ is the degree of the polynomial and $N$ and $p$ are large. Angularly, there are also $2p$ preferred sites. Other new akin crystallization phenomena take plake under less restricted probability hypotheses on the signal. A variety of results is described here, rigorously or on the basis of numerical experiments or asymptotic estimates.

D. Gaier: On the Relation Between En(f) and En(f'), 225--232
Let $G$ be a Jordan domain, let $f$ be holomorphic in $G$ and continuous on $\ol G$, and assume that also $f'$ has a continuous extension to $\ol G$. We investigate the relation between the errors of approximation by polynomials, $E_n(f)$ and $E_n(f')$, under certain assumptions on $G$. Similarly, we study this relation when $f$ and $f'$ are continuous on a Jordan arc.

F. W. Gehring, K. Hag: A Bound for Hyperbolic Distance in a Quasidisk, 233--240
A simply connected proper subdomain $D$ of the complex plane $\C$ is a $K$-quasidisk if and only if the hyperbolic distance $h_D(z_1,z_2)$ between $z_1$ and $z_2$ in $D$ is bounded above by $a\,j_D(z_1,z_2)+b$ where $a$ and $b$ are constants and $j_D$ a certain function of the ratios of the euclidean distance between $z_1$ and $z_2$ and the euclidean distances from $z_1$ and $z_2$ to $\partial D$. We derive here a distortion theorem for quasiconformal mappings which yields simple upper bounds for $a$ and $b$ as functions of $K$.

J. Godula, V. V. Starkov: On Regularity Theorems for Linearly Invariant Families of Analytic Functions in the Unit Polydisk, 241--258

R. Greiner: Generalized Jackson Kernels in Approximation Theory, 259--266
We study approximative identities
$$I_n(f)(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \chi_n(u) f(x-u) \,du$$
with positive even trigonometric polynomials $\chi_n$ of degree $n$ as kernels. It is well known that in general the approximants $I_n(f)$ are norm convergent to $f$ with order at most $n^{-2}$ and that convergence for the test function $f(x)=(2\sin(x/2))^2$ at $x=0$ implies norm convergence for a wide class of functions $f$.
We discuss the order of convergence at $x=0$ for $(2\sin(x/2))^{2r}$ where $r\geq 1$ is an integer and characterize kernels $\chi_n$ which produce best convergence. This leads to estimates of the type
$$\int_{-\pi}^{\pi} \chi_n(u) |u|^{2r} \,du = \go(n^{-2r})$$
which arise \eg in the investigation of the degree of approximation of differentiable functions. Generalized Jackson kernels will be shown to have the same (best possible) rate of convergence. All the kernels discussed are suitable for approximative identities. Moreover, their special properties find applications, also in complex approximation theory.

R. Grothmann: On the Problem of Poreda, 267--274
We investigate a problem posed by Poreda on the behaviour of the strong uniqueness constant with increased polynomial degree. It has been conjectured by Bartelt and McLaughlin that this constant tends to zero for all non-polynomial functions. In this paper, we give evidence for this and prove a special result, which we conjecture to be a worst case result.

M. Ito, M. Shiba: Area Theorems for Conformal Mapping and Rankine Ovoids, 275--284
We study the Rankine ovoids in hydrodynamics in connection with the theory of conformal mapping of the unit disk. We show that for any given complex number $\kappa$ with $|\kappa| < 1$ there is an essentially unique Rankine ovoid outside of which is precisely the image of a conformal mapping of the open unit disk with the expansion $1/z + \kappa z + o(|z|)$ near $z = 0$. The area of the Rankine ovoid is computed and compared with the maximum value of the possible area for the parameter $\kappa$ which is attained by an ellipse.

G. Jank, G. Kun: Solutions of Generalized Riccati Differential Equations and Their Approximation, 285--302
We study the solution of generalized matrix Riccati differential equations as they appear in differential games theory with various information structures. We propose a numerical approximation for the generalized Riccati differential equation arising from an optimally controlled differential game with memoryless feedback. This method makes use of the determination of the exact solution of associated nonsymmetric Riccati differential equations. Furthermore, we present an example of a generalized algebraic Riccati equation having several stabilizing positive semidefinite solutions.

E. A. Karatsuba: Fast Evaluation of Hypergeometric Functions by FEE, 303--314
An algorithm for fast calculation of hypergeometric functions for algebraic values of argument and parameters is proposed. The computational complexity is near to optimal. The merit of the method is the possibility of its implementation in parallel processing.

E. S. Katsoprinakis: On the Complex Rolle Set of a Polynomial, 315--326
Let $p(z)$ be a complex polynomial of degree $n \geq 2$ with zeros $z_j$ and let $w_k$ be the zeros of its derivative. If $z_1+ \cdots +z_n=0,$ then I. J. Schoenberg called the set $E=\{z_j, w_k\}$ the complex Rolle set" of $p(z)$ and conjectured that:
$$|w_1|^2 + \cdots + |w_{n-1}|^2 \leq {\frac{n-2}{n}} (|z_1|^2 + \cdots + |z_n|^2).$$
Here we verify special cases of this conjecture, using a generalization of Van den Berg's Theorem and the Theory of Majorization. Moreover, we arrive at a new conjecture, which implies the above conjecture and also another Schoenberg-type conjecture proposed by M.~G.~de Bruin and A.~Sharma.

C. A. Kokkinos: A Unified Orthonormalization Method for the Approximate Conformal Mapping of Simply and Multiply-Connected Domains, 327--344
We present a unified method for computing the following conformal maps: (a) The mapping of a finite $N$-connected domain bounded by $N\geq2$ ($N\geq1$) closed piecewise analytic Jordan curves onto a circular ring (disc) slit along $N-2$ ($N-1$) concentric circular arcs. (b) The mapping of a domain exterior to $N\geq1$ closed piecewise analytic Jordan curves onto the exterior of a circle (the whole complex plane) slit along $N-1$ ($N$) concentric circular arcs.

S. Koumandos: Positive Trigonometric Sums in the Theory of Univalent Functions, 345--358
We give several positive sine sums associated with certain problems on the logarithmic coefficients of univalent functions. In particular, we give a class of positive convex sine sums.

P. Kravanja, M. Van Barel, A. Haegemans: On Computing Zeros and Poles of Meromorphic Functions, 359--370
Given a meromorphic function~$f$, we present an accurate numerical method that computes all the zeros and poles of~$f$ that lie inside a Jordan curve~$\gamma$, together with their respective multiplicities and orders. An upper bound for the total number of poles of~$f$ that lie inside~$\gamma$ is assumed to be known. Our algorithm is based on numerical integration along~$\gamma$ and formal orthogonal polynomials. It uses the logarithmic derivative~$f'/f$. Initial approximations for the zeros and poles are not needed. Numerical examples illustrate the effectiveness of our approach.

E. (A. L.) Levin: Fast Decreasing Rational Functions and Their Applications, 371--386
Let $r_n$ be a rational function of the form $p_n(x)/p_n(-x)$, where $p_n$ is a polynomial of degree $n$. Let $\varphi$ be continuous and increasing on [0,1], with $\varphi(0) = 0$. Under what conditions on $\varphi$, does~a sequence $r_n$, $n\ge 1$, exist such that
$$|r_n(x)| \le C\exp \big(-n\varphi(x)\big),\quad x \in [0,1]\?$$
We present a survey of recent results concerning this problem. Some applications to rational approximation with locally geometric rates are also given.

W. Majchrzak: Harmonic Univalent Mappings of the Unit Disc onto a Vertical Strip, 387--396
In this paper the class ${\cal P}_H$ of harmonic, univalent, sense-preserving and normalized mappings of the unit disc onto a vertical strip is considered, as well its closure ${\cal F}_H$. An integral representation for functions in ${\cal F}_H$ is determined and several important properties are studied. In particular, the coefficient problem is partially solved.

D. Minda: Euclidean Circles of Curvature for Geodesics of Conformal Metrics, 397--404
The purpose of this note is to establish a simple property of euclidean circles of curvature for geodesic arcs relative to a conformal metric. For the special case of the hyperbolic metric on a Nehari region which is not M\"obius equivalent to a strip, this property gives an elegant geometrical interpretation for the Ahlfors-Weill quasiconformal reflection in the boundary of the region. The interpretation reveals that Ahlfors-Weill reflection is a natural extension of the classical notion of Schwarz reflection in a circle or line.

A. H. M. Murid, M. Z. Nashed, M. R. M. Razali: Some Integral Equations Related to the Riemann Map, 405--420
Each of the Szeg\"o and the Bergman kernel functions is known to satisfy a certain integral equation of the second kind. In this paper, based on a certain boundary relationship, an integral equation is formulated such that both the integral equations for the Szeg\"o and the Bergman kernels can be derived from that equation. New integral equations related to the Riemann map are also derived.

V. Nestoridis: An Extension of the Notion of Universal Taylor Series, 421--430
Let $\Omega$ be a simply connected domain $\Omega\subset{\Bbb C}$, $\Omega\neq {\Bbb C}$ and $\zeta\in\Omega$. For $f\in H(\Omega)$ we denote by $S_N(f,\zeta)(z)$ the sequence of partial sums of the Taylor development of $f$ with center $\zeta$. We prove that generically for $f\in H(\Omega)$ the following holds:
For every compact set $K$, $K\cap\Omega=\emptyset$ with $K^c$ connected and every function: $\phi:K\to{\Bbb C}$ continuous on $K$ and holomorphic in $K^o$, there exists a strictly increasing sequence $\lamda_n\in\{0,1,2,\ldots\}$, such that, for every compact set $L\subset\Omega$ we have
$$\sup_{\zeta\in L}\sup_{z\in K}|S_{\la_n}(f,\zeta)(z)-\phi(z)|\ra 0,\ \mbox{ as }n\ra+\infty.$$

M. Obradovic: Starlikeness of Certain Integral Transforms, 431--436
By using the method of differential-subordination we consider starlikeness of certain integral transforms in the unit disc.

I. E. Pritsker, R. S. Varga: Rational Approximation with Varying Weights in the Complex Plane, 437--448
Given an open bounded set $G$ in the complex plane and a weight function $W(z)$ which is analytic and different from zero in $G$, we consider the problem of locally uniform rational approximation of any function $f(z)$, which is analytic in $G$, by particular ray sequences of weighted rational functions of the form $W^{m+n} (z)R_{m,n}(z)$, where $R_{m,n}(z)=P_m(z)/Q_n(z),$ with $\deg P_{m} \leq m$ and $\deg Q_{n} \leq n.$ The main result of this paper is a necessary and sufficient condition for such an approximation to be valid. We also consider a number of applications of this result to various classical weights, and find explicit criteria for the possibility of weighted approximation in these cases.

D. V. Prokhorov: Radii of Neighborhoods for Coefficient Estimates of Functions Close to the Identity, 449--460
We estimate from below the radius $M(\alpha ,\beta )$ of a neighborhood $S^M$ of the identity function, where the Pick functions maximize the linear functional $I(\alpha ,\beta ;f)=\Re \{ a_4+\alpha a_3+\beta a_2\}$ for functions $f(z)=z+a_2z^2+\dots$ holomorphic and univalent in the unit disk and such that $|f(z)| < M$.

O. Roth: A Remark on the Loewner Differential Equation, 461--470
We give a control-theoretic proof of Pommerenke's result on the parametric representation of normalized univalent functions in the unit disk as solutions of the Loewner differential equation. The method consists in combining a classical result on finite-dimensional control-linear systems with Montel's theorem on normal families.

C. Rudälv: Repeated Bivariate Rational Approximants with Prescribed Poles, 471--484
Recently Walsh considered approximation to analytic functions on compact sets by interpolation with rational functions having prescribed poles. Ambroladze and Wallin studied this interpolation problem from a somewhat different point of view. Inspired by their ideas and by the approximants defined by Chaffy-Camus and Guillaume we construct rational approximants with prescribed poles to bivariate functions using a repeated interpolation procedure. In this paper we give some convergence results and numerical examples.

G. Schmeisser: Real Zeros of Bernstein Polynomials, 485--496
We describe subsets ${\cal U}$ of the real line with the property that, if a polynomial $f$ has all its zeros in ${\cal U}$, then so have all the Bernstein polynomials $B_n[f]$ of positive degree. Certain related questions including extensions to entire functions are also studied. Finally, we outline a connection with a recent result on polynomial transformations.

G. Schmieder: Location of Critical Points for Polynomials, 497--504
Let some complex polynomial be given in terms of its zeros. We estimate the location of the zeros of the derivative.

F. Stenger, R. Schmidtlein: Conformal Maps via Sinc Methods, 505--550
We derive a {\em Sinc procedure} for the construction of a conformal map, $f$, of a simply connected domain, or Riemann surface $B$ in the complex plane to the unit disc $U$. The construction is based on the solution of a boundary integral equation which always has a unique solution. We assume that ${\partial B}$, the boundary of $B$, consists of a finite number of analytic arcs, with well defined angles at the junctions. We also give an explicit procedure for evaluating $f$ in the interior of $B$. We furthermore give a brief description of an explicit {\em Sinc construction} which enables the computation of the inverse map $F = f^{-1}$ from $U$ to $B$, based on the computed $f$ on ${\partial B}$. Given any $\varepsilon > 0$, the time complexity of sequential computation of $f_\varepsilon$ on $\partial B$ such that $\sup_{\zeta \in {\partial B}}|f(\zeta) - f_\varepsilon(\zeta)| < \varepsilon$ is ${\cal O}\left((\log(\varepsilon))^6\right)$.

K. Stephenson: The Approximation of Conformal Structures via Circle Packing, 551--582
This is a pictorial tour and survey of circle packing techniques in the approximation of classical conformal objects. It begins with numerical conformal mapping and the conjecture of Thurston which launched this topic, moves to approximation of more general analytic functions, and ends with recent work on the approximation of conformal tilings and conformal structures.

E. Wegert: Nonlinear Riemann-Hilbert Problems -- History and Perspectives, 583--616
This contribution is devoted to a nonlinear boundary value problem for holomorphic functions in the complex unit disc which can be traced back to Bernhard Riemann. We do not intend to give an exhaustive survey of the problem; rather, the paper is meant as a shop window that presents several selected aspects. In order to give the reader a flavour of the subject we comment on the history, sketch some basic results, discuss a number of applications, indicate interrelations to other questions in complex analysis, and address a few open problems.

R. Wegmann: Two Plane Free Boundary Problems with Surface Tension, 617--634
We study the flow around a bubble and along a wall with~a gap. In both problems a free surface is balanced by pressure forces and surface tension. The flow covers a region which is described by an analytic function $f$ in the exterior of the unit disc. The equilibrium condition yields an equation for the derivative $f'$ which may be interpreted as an equation in a Hoelder space. This equation depends on two nondimensional parameters which measure the pressure difference and the surface tension. The implicit function theorem gives the existence of solutions for bubbles with sufficiently large inside pressure. For flow protruding through a gap solutions exist when the surface tension is large enough and the pressure difference is small. In these cases the Newton method converges. In each step of the Newton iteration a function theoretic boundary problem has to be solved. This can be used for numerical calculations. We present some numerical results.