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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 $N=4$.
- 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 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 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 $K$.
-
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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
$$\sup_{\zeta\in L}\sup_{z\in K}|S_{\la_n}(f,\zeta)(z)-\phi(z)|\ra 0,\
\mbox{ as }n\ra+\infty.$$
- By using the method of differential-subordination we
consider starlikeness of certain integral transforms in the unit disc.
- 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.
- 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 $|f(z)| < M$.
- 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.
- 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.
- 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.
- Let some complex polynomial be given in terms of its zeros.
We estimate the location of the zeros of the derivative.
- 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 ${\cal O}\left((\log(\varepsilon))^6\right)$.
- 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.
- 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.
- 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.