Computational Methods and Function Theory
Proceedings 1994
Abstracts
B. Aebischer:
Stable Convergence of Sequences of Möbius Transformations, 001--022
- The search for conditions which guarantee convergence of a parameter
depending continued fraction in a neighborhood of a parameter value
leads to a natural notion of stable convergence. Assume the sequence
of linear fractional transformations $S_n=s_1\circ\dots\circ s_n$,
$s_n(x) = \frac{a_n x+b_n}{c_n x+d_n}$, converges at some point
$x_0\in \hat{\Bbb C}$. (For continued fractions $x_0=0$, $s_n(x) =
\frac{a_n}{b_n+x}$.) Roughly, the convergence is called stable if for
any choice of linear fractional transformations $\tilde s_n$
sufficiently close to $s_n$, the sequence $\tilde S_n(x_0)$
converges, where $\tilde S_n = \tilde s_1\circ\dots\circ \tilde s_n$.
For instance, a C-fraction ${\bf K}(\frac{a_n z^{\alpha_n}}{1})$
($a_n \not=0$) which converges stably at $z_0 \in {\Bbb
C}\setminus\{0\}$ will converge in a neighborhood of $z_0$, provided
$\alpha_n$ is bounded. Similar assertions hold for general
T-fractions, associated continued fractions, and J-fractions.
If a subset of the M\"obius group satisfies a certain contraction
condition, then every sequence generated by elements of the set
converges stably. This result has many consequences, especially for
limit periodic continued fractions. Other applications include
stability versions of the theorem of Prings\-heim, the theorem of
Worpitzky, and the uniform parabola theorem. A weaker version of
stability is shown to imply stable convergence provided the $s_n$ are
contained in a compact subset of the M\"obius group.
The results given (as well as the methods used) also hold for M\"obius
transformations in higher dimension.
- For $0\leq\rho < 1$ let $\Omega _\rho $ be the parabolic region
$\Omega _\rho =\{u+iv\,:\,v^2\leq 4 (1-\rho )(u-\rho
)\}=\{w\,:\,|w-1|\leq1-2\rho +\RE w\}$. The class of normalized
analytic functions $f$ in the unit disk $U$ for which
$f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and $zf'(z)/f(z)\in\Omega _\rho ,\
z\in U$, is denoted by $S_p(\rho )$ and is called the class of
parabolic starlike functions of order $\rho $. For $\rho
=\frac{1}{2}$ this class of functions arises in the study of
uniformly covex functions introduced by Goodman and studied in detail
by W. Ma \& D. Minda, and F. R\o nning.
In the present paper we
obtain sharp upper bounds on the coefficients $|a_n|$, $n=2,3$ and
$4$ and the first four coefficients of functions inverse to
$f\in\Sp$. Unlike in the case of $\rho =\frac{1}{2}$, in the present
situation there are at least two extremal functions, and for the
fourth coefficient of the inverse functions, three different extremal
functions arise. If $\frac{zf'(z)}{f(z)}=1+\sum_{k=1}^{\infty}b_kz^k$
and $f\in\Sp$, we also obtain a sharp estimate on
$\sum_{k=1}^{\infty}|b_k|^2$, which yields a general Littlewood type
bound on $|a_n|$.
- Let $M_{n, p}(A, B, \alpha)$ denote the class of functions
$f$ of the form
$f(z) = z^p + \sum_{k=1}^\infty a_{p+k}z^{p+k}$ which are regular in the
unit disc $\D = \{z:\,|z| < 1\}$ and satisfy the condition
$$
\frac{{\cal D}^{n+p}f(z)}{z^p} \prec
\frac{1+[B+(A-B)(1-\frac{\alpha}{p})]z}{1+Bz},
\quad z \in \D,
$$
where $-1 \leq B < A \leq 1$, $0 \leq \alpha < p$ and $n$ is an integer
such that $n > -p$ and
$$
{\cal D}^{n+p-1}f(z) =
\frac{z^p(z^{n-1}f(z))^{(n+p-1)}}{(n+p-1)!}.
$$
In this paper we show that the functions in $M_{n,p}(A, B, \alpha)$ are
$p$-valent. Then we obtain sharp coefficient estimates, maximization of
$| a_{p+2} - \mu a^2_{p+1}|$ and a closure theorem for the class
$M_{n,p}(A,B,\alpha)$. We also obtain a sufficient condition, in terms of
coefficients, for a function to be in $M_{n,p}(A,B,\alpha)$ when
$-1 \le B < 0$.
- A general result on families of analytic functions with a
common invariant domain is applied to extend previous results on the
convergence of infinite exponentials with arbitrary complex exponents.
- Let $D$ be a bounded convex domain and suppose that $f$ is an
analytic map of $D$ into itself with $|f'| < 1$. Then the diameter of
$f^n(D)$ converges to zero and we obtain upper and lower estimates on
the rate of this convergence. The methods used also give results in
other circumstances; for example, for contractions of a domain in
$n$-dimensional Euclidean space.
- For a continuous $2\pi$-periodic real-valued function
$f$, we investigate the asymptotic behavior of the zeros of the error
function of best $L^2$-approximation by trigonometric polynomials.
The method of proof can also be applied for $L^p$-approximation,
$p>1$, by algebraic polynomials.
- A region $\Omega$ in the complex plane is called a Bloch
region provided it does not contain arbitrarily large disks. We investigate
the behavior of the density $\lambda_\Omega(w)$ of the hyperbolic metric on
Bloch regions. A number of sharp inequalities involving $\lambda_\Omega(w)$
and its gradient are obtained from results for locally univalent Bloch
functions. The extremal region for all of the inequalities is a punctured
disk. In addition, we show that $1/\lambda_\Omega(w)$ satisfies a H\"older
condition of order $\beta$ for some (all) $\beta \in (0,1)$ if and only if
$\Omega$ is a Bloch region.
- The classical trigonometric inequalities of Young and
Fej\'er-Jackson are reviewed and extended in the context of current work
on sums of Gegenbauer polynomials.
- Let $\{\alpha_n\}$ be a sequence of points in the open unit disk such that
$\alpha_n\rightarrow\alpha$, $|\alpha| < 1$, and let $\mu$ be a positive
measure on the unit circle. Let $\{\varphi_n\}$ be orthogonal rational
functions obtained by orthogonalizing the base $\{B_n\}$, where
\begin{eqnarray*}
B_0=1,
\qquad
B_n=\prod_{m=1}^n\frac{\overline{\alpha_m}}{|\alpha_m|}
\left(\frac{\alpha_m-z}{1-\overline{\alpha_m}z}\right),
\quad
n=1,2,\ldots,
\end{eqnarray*}
$(\frac{\overline{\alpha_m}}{|\alpha_m|}=-1$ if $\alpha_m=0$). The
functions $\varphi_n(z)$ and their superstar transforms $\varphi_n^*(z)
=B_n(z)\overline{\varphi_n(1/\overline{z})}$ satisfy a recurrence
relation of the form
\begin{eqnarray*}
\varphi_n(z)
=\frac{\kappa_n}{\kappa_{n-1}}
\left[\varepsilon_n\frac{z-\alpha_{n-1}}
{1-\overline{\alpha_n}z}\varphi_{n-1}(z)+
\delta_n\frac{1-\overline{\alpha_{n-1}}z}
{1-\overline{\alpha_n}z}\varphi_{n-1}^*(z)
\right],
\end{eqnarray*}
for certain constants
$\kappa_n$, $\delta_n$, $\varepsilon_n$. It is shown that if
$$
\sum_{m=1}^\infty \left[\left|1+\frac{\overline{\alpha_m}}{|\alpha_m|}
\overline{\varepsilon_m}\right|+|\delta_m|
\right] < \infty,
$$
then $\{\varphi_n^*(z)\}$ converges uniformly on the closed unit disk to
the function
\begin{eqnarray*}
\pi_a(z)=
\frac{1}{\sqrt{2\pi}}e^{-i\lambda}
\frac{\sqrt{1-|\alpha|^2}}{(1-\overline{\alpha}z)}\,
\exp\!\left[-\frac{1}{4\pi}\,\int_{-\pi}^\pi
\frac{e^{i\theta}+z}{e^{i\theta}-z}\ln \mu'(\theta)\,d\theta
\right],
\quad
\lambda\in{\Bbb R}.
\end{eqnarray*}
The condition above reduces to the classical condition
$\sum_{m=1}^\infty|\delta_m| < \infty$ when $\alpha_n=0$ for all $n$.
- Bank, Laine and Langley showed that when a
solution of $y''+(e^z-K)y=0$ has a finite exponent of convergence on its
zero-sequence, then $K=(2n+1)^2/16$, where $n$ is a non-negative integer. We
give a refinement of their result by using a different approach, which
allows us to consider another application.
- Simultaneous rational approximants (SRA) to a pair of Markov
functions $(f_1,f_2)$ that forms a Nikishin system are investigated.
It is well known that all multi-indices in the table of SRAs are normal
and that all common denominators $Q_0$ satisfy a multiple orthogonality
relation. We show that all the $Q_0$'s also satisfy an ordinary orthogonality
relation, but now with respect to a varying measure. Error formulae are
presented and an interlacing property proved for the finite zeros of the
two error functions in the diagonal and quasi-diagonal cases.
- This work continues a program to generalize one variable
geometric function theory to several complex variables. The first
mappings studied are required to be locally biholomorphic on
spaces of matrices into ${\Bbb C}^n$. It is required that these
mappings have a limited rate of growth as the boundary of the
domain is approached. (Otherwise the Bloch constant may not
exist.) These requirements specify classes of mappings. For each
class of mappings there is a positive number $r$ such that, for each
mapping $f$ in the class, there is a subdomain which $f$ carries
onto a ball of radius $r$ in a one-to-one fashion. The supremum of
such numbers $r$ is the Bloch constant for the class of mappings.
For a wide range of classes, a positive lower bound is found for the
Bloch constant.\\
For the last portion of the paper, the mappings are required to be
holomorphic, not necessarily locally biholomorphic. For each class
of mappings, there is a positive number $\rho$ such that, for each
mapping $f$, there is a hyperbolic ball of radius $\rho$ in the
domain of $f$ on which $f$ is one-to-one. The supremum of
such numbers $\rho$ is the Marden constant for the class of
mappings. From the earlier results of the paper, a positive
lower bound of the Marden constant is obtained.
- We give a survey of old and new methods to determine the modulus $m(Q)$
of a quadrilateral $Q$ and the modulus $M(G)$ of a ring shaped domain $G$.
We concentrate on direct methods, \ie on methods which avoid the computation
of the corresponding conformal map.
- We show how the theory of holomorphic motions and estimates for
Schottky's Theorem can be used to extend a known inequality concerning two
generator discrete groups of M\"{o}bius transformations.
- Let $f$ be analytic and univalent in the unit disk
$\scriptstyle \Bbb D$ and maps it onto a Jordan domain in the
complex plane $\scriptstyle \Bbb C$. First we show that
$\int\!\!\!\int |\{f,z\}|dm_z < \infty$ implies that $ g=\ln f'$ is continuous
in $\overline{\scriptstyle \Bbb D}$. Then we give conditions for $ f'(z)$
or $1/f'(z)$ to belong to the Hardy space $ H^1$. Finally we note that
if $f$ be analytic in $\scriptstyle \Bbb D$ then
$\int\!\!\!\int |\{f,z\}|^{1/2}dm_z < \infty$ implies that $f$ is of bounded
characteristic.
- Some new inequalities for sums of Legendre polynomials are used to prove
the positivity of the Cotes numbers for the quadrature method of interpolating
at the zeros of Jacobi polynomials and then integrating with respect to
$w(x) \mbox{d}x$ on $[-1,1]$, where
$$
w(x) = (1-x)^c (1+x)^d,\ c > -1,\ d > -1 \quad (c \neq 0, d \neq 0).
$$
The positivity of some related sums of Jacobi polynomials is also established.
- Let $G \subsetneq {\Bbb C }$ be a simply-connected domain and let $\sg$
be its group of conformal automorphisms with the topology of {\it
uniform} chordal convergence on $G$.\\
In Gaier, D., \"Uber R\"aume konformer Selbstabbildungen ebener Gebiete,
{\sl Math.\ Z.\/} {\bf 187} (1984), 227--257,
Gaier raised the question whether each space $\sg$ is locally compact.
In this paper we give examples of simply-connected domains with exactly
one or two non-degenerate prime ends and {\it non}-locally compact
automorphism spaces.
- As simple applications of Szeg\H{o}'s theory of orthogonal polynomials,
we generalize some C.~K.~Chui's convergence results on double
least-squares inverse method that has been used to approximate a
polynomial by another polynomial having no zeros in the closed unit disk.
- In this paper we compare solutions $\{\tilde Z_n\}$ of a
recurrence relation
$$
\tilde Z_n+\tilde a_n^{(1)}\tilde Z_{n-1}+\tilde a_n^{(2)}\tilde Z_{n-2}=0;
\quad\tilde a_n^{(2)}\ne 0;\qquad n=1,2,3,\dots$$
to solutions $\{Z_n\}$ of the perturbed equation
$$Z_n+a_n^{(1)}Z_{n-1}+a_n^{(2)}Z_{n-2}=0;\quad a_n^{(2)}\ne 0;
\qquad n=1,2,3,\dots,$$
where $\lim(a_n^{(1)}-\tilde a_n^{(1)})=\lim(a_n^{(2)}-\tilde a_n^{(2)})=0$.
In particular we find sufficient conditions for
$$\lim_{n\to\infty}(Z_n/Z_{n-1}-\tilde Z_n/\tilde Z_{n-1})=0
$$
and
$$\lim_{n\to\infty}Z_n/\tilde Z_n=\alpha\qquad\mbox{for some}\,\,\,\alpha\in\Bbb
C.
$$
The results generalize classical results by Perron and Evgrafov.
- We are investigating the size of minimal polynomials and Faber polynomials
on annular sectors. The size is the uniform norm of the corresponding polynomials
on the underlying sector. If the size is less than one, the corresponding polynomial
is suitable for the use in so-called polynomial-based iteration-schemes for solving
linear systems. For intrinsic reasons all polynomials $p$ to be considered must
satisfy the normalization condition $p(0)=1$. The annular sectors play the
r\^ole of an inclusion set for the eigenvalues of the underlying matrix
in the linear system to be solved. The minimal polynomials are those
with least uniform norm in this class. The computation of minimal
polynomials requires a Remez-type algorithm for complex cases. Faber
polynomials are computed by means of Schwarz-Christoffel-formulae. Here
singular integrals of different types occur. Various Gauss-formulae with
high knot numbers are employed. The annular sectors have the remarkable
property, that they are invariant under monomial mappings, which also
allows the use of several unconnected sectors of the same size
distributed equiangularly over the annulus. This is of great help in
certain applications, where the mentioned eigenvalues are located in a
disconnected set. There are comparisons in graphical form of both types
of polynomials with respect to degree and to the different types of
sectors. Most of the numerical work is included in these figures.
- This paper is concerned with certain aspects of the theory and
application of a domain decomposition method for computing the conformal
modules of long quadrilaterals. Our main purpose is to make use of the
theory of the method in order to investigate the quality of certain
heuristic rules that we have come across in the VLSI literature, in
connection with the measurement of resistance values of integrated
circuit networks.
- We investigate the energy of arrangements of $N$ points
(charged particles) on the surface of the unit sphere in ${\Bbb R}^3$,
interacting through a power law potential $ V(r) = r^\alpha, \;\;
- 2 < \alpha < 2, \;\; \alpha \neq 0 $ and $ V (r) = \log (1/r)$ for
$ \alpha = 0 $. Results of numerical experiments are presented for the
three classical cases $ \alpha = 0 , \pm 1, \mbox{ for } N=2,\ldots,200$.
We also investigate the geometric structures of the equilibrium points
and their group properties. In the case $ \alpha = 0 $, we prove that the
$N$ points of minimal energy must be well-separated in the sense that
any two distinct points are at least $ (3/5) (1/\sqrt{N})$ units
distance apart.
- We consider a derivative estimate and a
growth estimate for entire functions of exponential type in the $L^p$
norm and various modifications and refinements. Then we describe
generalizations which include extensions to $p\in (0,1),$ the range
where $\|.\|_p$ is no longer a norm. Next we show that all of these
results can be deduced from one inequality, which is a growth estimate
with a free parameter. Finally we pose a problem.
- We investigate the classes of normalized functions $f(z)$
analytic in the open unit disk ${\Bbb D}=\{z:|z| < 1\}$ for which
$\frac{zf'(z)}{f(z)}$ or $\frac{zf''(z)}{f'(z)}+1$ ranges over the
parabolic region $\Omega=\{w:|w-1| < \Real w\}$. We also discuss
some integral operators and radius problems.
- Results about simultaneous rational approximants will be surveyed.
Special attention will be paid to the approximation of Angelesco and
Nikishin systems. Simultaneous rational approximants are a natural
generalization of Pad\'e approximants and continued
fractions. Instead of only one function now a vector
$(f_1,\dots,f_m)$ of $m$ functions is simultaneously approximated by
a vector $(q_1/q_0,\dots,q_m/q_0)$ of rational functions with a
common denominator $q_0$. It is well known that the theory of Pad\'e
approximants is closely related with orthogonal polynomials, three
term recurrence relations, and Gau\ss{} quadrature. In an analoguous
way the theory of simultaneous rational approximants is connected
with multiple orthogonal polynomials, $m+2$-term recurrence
relations, and simultaneous quadrature. After surveying the basic
concepts underlying the definition of simultaneous rational
approximants, we shortly review multiple orthogonal analogues to
classical orthogonal polynomials. We then turn to the approximation
of systems (vectors) of Markov functions. Contrary to the case of a
single Markov function, where Markov's Theorem gives a very complete
answer to the convergence problem, in the vector case general results
are comparatively rare; such results have been proved so far only for
special systems, namely the Angelesco and the Nikishin systems.
Results about Angelesco systems will be reviewed and new results
about Nikishin systems will be presented. In the proofs of
convergence for both types of systems an equilibrium problem for
vector potentials plays a key role. The connection between the
vector equilibrium problem and certain algebraic functions will be
discussed.