26. CJM 2003 (vol 55 pp. 576)
 Lukashov, A. L.; Peherstorfer, F.

Automorphic Orthogonal and Extremal Polynomials
It is well known that many polynomials which solve extremal problems
on a single interval as the Chebyshev or the BernsteinSzeg\"o
polynomials can be represented by trigonometric functions and their
inverses. On two intervals one has elliptic instead of trigonometric
functions. In this paper we show that the counterparts of the Chebyshev
and BernsteinSzeg\"o polynomials for several intervals can be represented
with the help of automorphic functions, socalled SchottkyBurnside
functions. Based on this representation and using the SchottkyBurnside
automorphic functions as a tool several extremal properties of such
polynomials as orthogonality properties, extremal properties with
respect to the maximum norm, behaviour of zeros and recurrence
coefficients {\it etc.} are derived.
Categories:42C05, 30F35, 31A15, 41A21, 41A50 

27. CJM 2003 (vol 55 pp. 379)
 Stessin, Michael; Zhu, Kehe

Generalized Factorization in Hardy Spaces and the Commutant of Toeplitz Operators
Every classical inner function $\varphi$ in the unit disk gives rise to
a certain factorization of functions in Hardy spaces. This factorization,
which we call the generalized Riesz factorization, coincides with the
classical Riesz factorization when $\varphi(z)=z$. In this paper we prove
several results about the generalized Riesz factorization, and we apply
this factorization theory to obtain a new description of the commutant
of analytic Toeplitz operators with inner symbols on a Hardy space. We
also discuss several related issues in the context of the Bergman space.
Categories:47B35, 30D55, 47A15 

28. CJM 2002 (vol 54 pp. 945)
 Boivin, André; Gauthier, Paul M.; Paramonov, Petr V.

Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications
Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jetdistributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
Keywords:approximation on closed sets, elliptic operator, strongly elliptic operator, $L$meromorphic and $L$analytic functions, localization operator, Banach space of distributions, Dirichlet problem Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30 

29. CJM 2002 (vol 54 pp. 709)
 Ismail, Mourad E. H.; Stanton, Dennis

$q$Integral and Moment Representations for $q$Orthogonal Polynomials
We develop a method for deriving integral representations of certain
orthogonal polynomials as moments. These moment representations are
applied to find linear and multilinear generating functions for
$q$orthogonal polynomials. As a byproduct we establish new
transformation formulas for combinations of basic hypergeometric
functions, including a new representation of the $q$exponential
function $\mathcal{E}_q$.
Keywords:$q$integral, $q$orthogonal polynomials, associated polynomials, $q$difference equations, generating functions, AlSalamChihara polynomials, continuous $q$ultraspherical polynomials Categories:33D45, 33D20, 33C45, 30E05 

30. CJM 2002 (vol 54 pp. 648)
 Yuan, Wenjun; Li, Yezhou

Rational Solutions of PainlevÃ© Equations
Consider the sixth Painlev\'e equation~(P$_6$) below where $\alpha$,
$\beta$, $\gamma$ and $\delta$ are complex parameters. We prove the
necessary and sufficient conditions for the existence of rational
solutions of equation~(P$_6$) in term of special relations among the
parameters. The number of distinct rational solutions in each case is
exactly one or two or infinite. And each of them may be generated by
means of transformation group found by Okamoto [7] and B\"acklund
transformations found by Fokas and Yortsos [4]. A list of rational
solutions is included in the appendix. For the sake of completeness,
we collected all the corresponding results of other five Painlev\'e
equations (P$_1$)(P$_5$) below, which have been investigated by many
authors [1][7].
Keywords:PainlevÃ© differential equation, rational function, BÃ¤cklund transformation Categories:30D35, 34A20 

31. CJM 2002 (vol 54 pp. 324)
 Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela

Parametric Representation of Univalent Mappings in Several Complex Variables
Let $B$ be the unit ball of $\bb{C}^n$ with respect to an arbitrary norm. We
prove that the analog of the Carath\'eodory set, {\it i.e.} the set of normalized
holomorphic mappings from $B$ into $\bb{C}^n$ of ``positive real part'', is
compact. This leads to improvements in the existence theorems for the Loewner
differential equation in several complex variables. We investigate a subset
of the normalized biholomorphic mappings of $B$ which arises in the study of
the Loewner equation, namely the set $S^0(B)$ of mappings which have
parametric representation. For the case of the unit polydisc these mappings
were studied by Poreda, and on the Euclidean unit ball they were studied by
Kohr. As in Kohr's work, we consider subsets of $S^0(B)$ obtained by placing
restrictions on the mapping from the Carath\'eodory set which occurs in the
Loewner equation. We obtain growth and covering theorems for these subsets of
$S^0(B)$ as well as coefficient estimates, and consider various examples.
Also we shall see that in higher dimensions there exist mappings in $S(B)$
which can be imbedded in Loewner chains, but which do not have parametric
representation.
Categories:32H02, 30C45 

32. CJM 2002 (vol 54 pp. 239)
 Cartwright, Donald I.; Steger, Tim

Elementary Symmetric Polynomials in Numbers of Modulus $1$
We describe the set of numbers $\sigma_k(z_1,\ldots,z_{n+1})$, where
$z_1,\ldots,z_{n+1}$ are complex numbers of modulus $1$ for which
$z_1z_2\cdots z_{n+1}=1$, and $\sigma_k$ denotes the $k$th
elementary symmetric polynomial. Consequently, we give sharp
constraints on the coefficients of a complex polynomial all of whose
roots are of the same modulus. Another application is the calculation
of the spectrum of certain adjacency operators arising naturally
on a building of type ${\tilde A}_n$.
Categories:05E05, 33C45, 30C15, 51E24 

33. CJM 2000 (vol 52 pp. 982)
 Lárusson, Finnur

Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds
Let $Y$ be an infinite covering space of a projective manifold
$M$ in $\P^N$ of dimension $n\geq 2$. Let $C$ be the intersection with
$M$ of at most $n1$ generic hypersurfaces of degree $d$ in $\mathbb{P}^N$.
The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$
be the smoothed distance from a fixed point in $Y$ in a metric pulled up
from $M$. Let $\O_\phi(X)$ be the Hilbert space of holomorphic
functions $f$ on $X$ such that $f^2 e^{\phi}$ is integrable on $X$, and
define $\O_\phi(Y)$ similarly. Our main result is that (under more
general hypotheses than described here) the restriction $\O_\phi(Y)
\to \O_\phi(X)$ is an isomorphism for $d$ large enough.
This yields new examples of Riemann surfaces and domains of holomorphy
in $\C^n$ with corona. We consider the important special case when $Y$
is the unit ball $\B$ in $\C^n$, and show that for $d$ large enough,
every bounded holomorphic function on $X$ extends to a unique function
in the intersection of all the nontrivial weighted Bergman spaces on
$\B$. Finally, assuming that the covering group is arithmetic, we
establish three dichotomies concerning the extension of bounded
holomorphic and harmonic functions from $X$ to $\B$.
Categories:32A10, 14E20, 30F99, 32M15 

34. CJM 2000 (vol 52 pp. 815)
 Lubinsky, D. S.

On the Maximum and Minimum Modulus of Rational Functions
We show that if $m$, $n\geq 0$, $\lambda >1$, and $R$ is a rational function
with numerator, denominator of degree $\leq m$, $n$, respectively, then there
exists a set $\mathcal{S}\subset [0,1] $ of linear measure $\geq
\frac{1}{4}\exp (\frac{13}{\log \lambda })$ such that for $r\in
\mathcal{S}$,
\[
\max_{z =r} R(z) / \min_{z =r}  R(z) \leq \lambda ^{m+n}.
\]
Here, one may not replace $\frac{1}{4}\exp ( \frac{13}{\log \lambda })$
by $\exp (\frac{2\varepsilon }{\log \lambda })$, for any $\varepsilon >0$.
As our motivating application, we prove a convergence result for diagonal
Pad\'{e} approximants for functions meromorphic in the unit ball.
Categories:30E10, 30C15, 31A15, 41A21 

35. CJM 1999 (vol 51 pp. 915)
 Balogh, Zoltán M.; Leuenberger, Christoph

Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers
Consider the polynomial hull of a smoothly varying family of
strictly convex smooth domains fibered over the unit circle. It is
wellknown that the boundary of the hull is foliated by graphs of
analytic discs. We prove that this foliation is smooth, and we
show that it induces a complex flow of contactomorphisms. These
mappings are quasiconformal in the sense of Kor\'anyi and Reimann.
A similar bound on their quasiconformal distortion holds as in the
onedimensional case of holomorphic motions. The special case when
the fibers are rotations of a fixed domain in $\C^2$ is studied in
details.
Categories:32E20, 30C65 

36. CJM 1999 (vol 51 pp. 977)
 Fisher, Stephen D.; Khavinson, Dmitry

Extreme PickNevanlinna Interpolants
Following the investigations of B.~Abrahamse [1], F.~Forelli [11],
M.~Heins [14] and others, we continue the study of the
PickNevanlinna interpolation problem in multiplyconnected planar
domains. One major focus is on the problem of characterizing the
extreme points of the convex set of interpolants of a fixed data
set. Several other related problems are discussed.
Categories:30D50, 30D99 

37. CJM 1999 (vol 51 pp. 470)
 Bshouty, D.; Hengartner, W.

Exterior Univalent Harmonic Mappings With Finite Blaschke Dilatations
In this article we characterize the univalent harmonic mappings from
the exterior of the unit disk, $\Delta$, onto a simply connected
domain $\Omega$ containing infinity and which are solutions of the system
of elliptic partial differential equations $\fzbb = a(z)f_z(z)$
where the second dilatation function $a(z)$ is a finite Blaschke
product. At the end of this article, we apply our results to
nonparametric minimal surfaces having the property that the image
of its Gauss map is the upper halfsphere covered once or twice.
Keywords:harmonic mappings, minimal surfaces Categories:30C55, 30C62, 49Q05 

38. CJM 1999 (vol 51 pp. 147)
 Suárez, Daniel

Homeomorphic Analytic Maps into the Maximal Ideal Space of $H^\infty$
Let $m$ be a point of the maximal ideal space of $\papa$ with
nontrivial Gleason part $P(m)$. If $L_m \colon \disc \rr P(m)$ is the
Hoffman map, we show that $\papa \circ L_m$ is a closed subalgebra
of $\papa$. We characterize the points $m$ for which $L_m$ is a
homeomorphism in terms of interpolating sequences, and we show that in
this case $\papa \circ L_m$ coincides with $\papa$. Also, if
$I_m$ is the ideal of functions in $\papa$ that identically vanish
on $P(m)$, we estimate the distance of any $f\in \papa$ to $I_m$.
Categories:30H05, 46J20 

39. CJM 1999 (vol 51 pp. 117)
40. CJM 1998 (vol 50 pp. 547)
41. CJM 1998 (vol 50 pp. 449)
42. CJM 1998 (vol 50 pp. 620)
43. CJM 1998 (vol 50 pp. 595)
 Luo, Donghan; MacGregor, Thomas

Multipliers of fractional Cauchy transforms and smoothness conditions
This paper studies conditions on an analytic function that imply it
belongs to ${\cal M}_\alpha$, the set of multipliers of the family of
functions given by $f(z) = \int_{\zeta=1} {1 \over
(1\overline\zeta z)^\alpha} \,d\mu (\zeta)$ $(z<1)$ where $\mu$ is a
complex Borel measure on the unit circle and $\alpha >0$. There are
two main theorems. The first asserts that if $0<\alpha<1$ and
$\sup_{\zeta=1} \int^1_0 f'(r\zeta) (1r)^{\alpha1} \,dr<\infty$
then $f \in {\cal M}_\alpha$. The second asserts that if $0<\alpha
\leq 1$, $f \in H^\infty$ and $\sup_t \int^\pi_0 {f(e^{i(t+s)}) 
2f(e^{it}) + f(e^{i(ts)}) \over s^{2\alpha}} \, ds < \infty$ then
$f \in {\cal M}_\alpha$. The conditions in these theorems are shown
to relate to a number of smoothness conditions on the unit circle for
a function analytic in the open unit disk and continuous in its closure.
Categories:30E20, 30D50 

44. CJM 1997 (vol 49 pp. 887)
 Borwein, Peter; Pinner, Christopher

Polynomials with $\{ 0, +1, 1\}$ coefficients and a root close to a given point
For a fixed algebraic number $\alpha$ we
discuss how closely $\alpha$ can be approximated by
a root of a $\{0,+1,1\}$ polynomial of given degree.
We show that the worst rate of approximation tends to
occur for roots of unity, particularly those of small degree.
For roots of unity these bounds depend on
the order of vanishing, $k$, of the polynomial at $\alpha$.
In particular we obtain the following. Let
${\cal B}_{N}$ denote the set of roots of all
$\{0,+1,1\}$ polynomials of degree at most $N$ and
${\cal B}_{N}(\alpha,k)$ the roots of those
polynomials that have a root of order at most $k$
at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$
we show that
\[
\min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} \alpha
\beta \asymp \frac{1}{\alpha^{N}},
\]
and for a root of unity $\alpha$ that
\[
\min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}}
\alpha \beta\asymp \frac{1}{N^{(k+1) \left\lceil
\frac{1}{2}\phi (d)\right\rceil +1}}.
\]
We study in detail the case of $\alpha=1$, where, by far, the
best approximations are real.
We give fairly precise bounds on the closest real root to 1.
When $k=0$ or 1 we
can describe the extremal polynomials explicitly.
Keywords:Mahler measure, zero one polynomials, Pisot numbers, root separation Categories:11J68, 30C10 

45. CJM 1997 (vol 49 pp. 617)
 Stahl, Saul

On the zeros of some genus polynomials
In the genus polynomial of the graph $G$, the coefficient of $x^k$
is the number of distinct embeddings of the graph $G$ on the
oriented surface of genus $k$. It is shown that for several
infinite families of graphs all the zeros of the genus polynomial
are real and negative. This implies that their coefficients, which
constitute the genus distribution of the graph, are log concave and
therefore also unimodal. The geometric distribution of the zeros
of some of these polynomials is also investigated and some new
genus polynomials are presented.
Categories:05C10, 05A15, 30C15, 26C10 

46. CJM 1997 (vol 49 pp. 520)
 Ismail, Mourad E. H.; Stanton, Dennis

Classical orthogonal polynomials as moments
We show that the Meixner, Pollaczek, MeixnerPollaczek, the continuous
$q$ultraspherical polynomials and AlSalamChihara polynomials, in
certain normalization, are moments of probability measures. We use
this fact to derive bilinear and multilinear generating functions for
some of these polynomials. We also comment on the corresponding formulas
for the Charlier, Hermite and Laguerre polynomials.
Keywords:Classical orthogonal polynomials, \ACP, continuous, $q$ultraspherical polynomials, generating functions, multilinear, generating functions, transformation formulas, umbral calculus Categories:33D45, 33D20, 33C45, 30E05 

47. CJM 1997 (vol 49 pp. 100)
 Lance, T. L.; Stessin, M. I.

Multiplication Invariant Subspaces of Hardy Spaces
This paper studies closed subspaces $L$
of the Hardy spaces $H^p$ which are $g$invariant ({\it i.e.},
$g\cdot L \subseteq L)$ where $g$ is inner, $g\neq 1$. If
$p=2$, the Wold decomposition theorem implies that there is
a countable ``$g$basis'' $f_1, f_2,\ldots$ of
$L$ in the sense that $L$ is a direct sum of spaces
$f_j\cdot H^2[g]$ where $H^2[g] = \{f\circ g \mid f\in H^2\}$.
The basis elements $f_j$ satisfy the
additional property that $\int_T f_j^2 g^k=0$,
$k=1,2,\ldots\,.$ We call such functions $g$$2$inner.
It also
follows that any $f\in H^2$ can be factored $f=h_{f,2}\cdot
(F_2\circ g)$ where $h_{f,2}$ is $g$$2$inner and $F$ is
outer, generalizing the classical Riesz factorization.
Using $L^p$ estimates for the canonical decomposition of
$H^2$, we find a factorization $f=h_{f,p} \cdot (F_p \circ
g)$ for $f\in H^p$. If $p\geq 1$ and $g$ is a finite
Blaschke product we obtain, for any $g$invariant
$L\subseteq H^p$, a finite $g$basis of $g$$p$inner
functions.
Categories:30H05, 46E15, 47B38 

48. CJM 1997 (vol 49 pp. 55)
 Chen, Huaihui; Gauthier, Paul M.

Normal Functions: $L^p$ Estimates
For a meromorphic (or harmonic) function $f$, let us call the dilation
of $f$ at $z$ the ratio of the (spherical) metric at $f(z)$ and the
(hyperbolic) metric at $z$. Inequalities are known which estimate
the $\sup$ norm of the dilation in terms of its $L^p$ norm, for $p>2$,
while capitalizing on the symmetries of $f$. In the present paper
we weaken the hypothesis by showing that such estimates persist
even if the $L^p$ norms are taken only over the set of $z$ on which
$f$ takes values in a fixed spherical disk. Naturally, the bigger
the disk, the better the estimate. Also, We give estimates for
holomorphic functions without zeros and for harmonic functions in
the case that $p=2$.
Categories:30D45, 30F35 
