Expand all Collapse all | Results 1 - 25 of 44 |
1. CMB Online first
On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions We recently introduced a weighted Banach algebra $\mathfrak{A}_G^n$ of
functions which are holomorphic on the unit disc $\mathbb{D}$, continuous
up to the boundary and of the class $C^{(n)}$ at all points where
the function $G$ does not vanish. Here, $G$ refers to a function
of the disc algebra without zeros on $\mathbb{D}$. Then we proved that
all closed ideals in $\mathfrak{A}_G^n$ with at most countable hull are
standard. In the present paper, on the assumption that $G$ is
an outer function in $C^{(n)}(\overline{\mathbb{D}})$ having infinite roots
in $\mathfrak{A}_G^n$ and countable zero set $h(G)$, we show that all the
closed ideals $I$ with hull containing $h(G)$ are standard.
Keywords:Banach algebra, disc algebra, holomorphic spaces, standard ideal Categories:46J15, 46J20, 30H50 |
2. CMB Online first
Schwarz Lemma at the Boundary of the Egg Domain $B_{p_1, p_2}$ in $\mathbb{C}^n$ Let $B_{p_1, p_2}=\{z\in\mathbb{C}^n:
|z_1|^{p_1}+|z_2|^{p_2}+\cdots+|z_n|^{p_2}\lt 1\}$
be an egg domain in $\mathbb{C}^n$. In this paper, we first
characterize the Kobayashi metric on $B_{p_1, p_2}\,(p_1\geq
1, p_2\geq 1)$,
and then establish a new type of the classical boundary Schwarz
lemma at $z_0\in\partial{B_{p_1, p_2}}$ for holomorphic self-mappings
of $B_{p_1, p_2}(p_1\geq 1, p_2\gt 1)$, where $z_0=(e^{i\theta},
0, \dots, 0)'$ and $\theta\in \mathbb{R}$.
Keywords:holomorphic mapping, Schwarz lemma, Kobayashi metric, egg domain Categories:32H02, 30C80, 32A30 |
3. CMB 2013 (vol 57 pp. 870)
A Short Note on Short Pants It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and SeppÃ¤lÃ¤. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.
Keywords:hyperbolic surfaces, geodesics, pants decompositions Categories:30F10, 32G15, 53C22 |
4. CMB 2012 (vol 56 pp. 881)
Free Groups Generated by Two Heisenberg Translations In this paper, we will discuss the groups generated by two
Heisenberg translations of $\mathbf{PU}(2,1)$ and determine when they are free.
Keywords:free group, Heisenberg group, complex triangle group Categories:30F40, 22E40, 20H10 |
5. CMB 2012 (vol 57 pp. 80)
Semicrossed Products of the Disk Algebra and the Jacobson Radical We consider semicrossed products of the disk algebra with respect to
endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical
of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic,
we show that the semicrossed product contains no nonzero quasinilpotent
elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step,
the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.
Keywords:semicrossed product, disk algebra, Jacobson radical Categories:47L65, 47L20, 30J10, 30H50 |
6. CMB 2012 (vol 56 pp. 466)
Inclusion Relations for New Function Spaces on Riemann Surfaces We introduce and study some new function spaces on Riemann
surfaces. For certain parameter values these spaces coincide with
the classical Dirichlet space, BMOA or the recently
defined $Q_p$ space. We establish inclusion relations that
generalize earlier known inclusions between the above-mentioned
spaces.
Keywords:Bloch space, BMOA, $Q_p$, Green's function, hyperbolic Riemann surface Categories:30F35, 30H25, 30H30 |
7. CMB 2012 (vol 56 pp. 769)
A Non-zero Value Shared by an Entire Function and its Linear Differential Polynomials In this paper we study uniqueness of entire functions
sharing a non-zero finite value with linear differential polynomials
and address a result of W. Wang and P. Li.
Keywords:entire function, linear differential polynomial, value sharing Category:30D35 |
8. CMB 2012 (vol 56 pp. 544)
Universally Overconvergent Power Series via the Riemann Zeta-function The Riemann zeta-function is employed to generate universally overconvergent power series.
Keywords:overconvergence, zeta-function Categories:30K05, 11M06 |
9. CMB 2012 (vol 56 pp. 241)
Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius We prove variants of Schwarz's lemma involving monotonicity
properties of condenser capacity and inner radius. Also, we
examine when a similar monotonicity property holds for the
hyperbolic metric.
Keywords:condenser capacity, inner radius, hyperbolic metric, Schwarz's lemma Categories:30C80, 30F45, 31A15 |
10. CMB 2011 (vol 56 pp. 229)
CesÃ ro Operators on the Hardy Spaces of the Half-Plane In this article we study the CesÃ ro
operator
$$
\mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta,
$$
and its companion operator $\mathcal{T}$ on Hardy spaces of the
upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$ as
resolvents for appropriate semigroups of composition operators and we
find the norm and the spectrum in each case. The relation of
$\mathcal{C}$ and $\mathcal{T}$ with the corresponding Ces\`{a}ro
operators on Lebesgue spaces $L^p(\mathbb R)$ of the boundary line is also
discussed.
Keywords:CesÃ ro operators, Hardy spaces, semigroups, composition operators Categories:47B38, 30H10, 47D03 |
11. CMB 2011 (vol 56 pp. 194)
On the Smallest and Largest Zeros of MÃ¼ntz-Legendre Polynomials MÃ¼ntz-Legendre
polynomials $L_n(\Lambda;x)$ associated with a
sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the
system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in
$L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s
are distinct, it is well known that $L_n(\Lambda;x)$ has exactly $n$
zeros $l_{n,n}\lt l_{n-1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ on $(0,1)$.
First we prove the following global bound for the smallest zero,
$$
\exp\biggl(-4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}.
$$
An important consequence is that if the associated MÃ¼ntz space is
non-dense in $L_2[0,1]$, then
$$
\inf_{n}x_{n,n}\geq
\exp\biggl({-4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0,
$$
so
the elements $L_n(\Lambda;x)$ have no zeros close to 0.
Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed,
$$
\lim_{n\rightarrow\infty} \vert \log l_{k,n}\vert \sum_{j=0}^n
(2\lambda_j+1)= \Bigl(\frac{j_k}{2}\Bigr)^2,
$$
where $j_k$ denotes the $k$-th zero of the Bessel function $J_0$.
Keywords:MÃ¼ntz polynomials, MÃ¼ntz-Legendre polynomials Categories:42C05, 42C99, 41A60, 30B50 |
12. CMB 2011 (vol 55 pp. 509)
Domains of Injective Holomorphy A domain $\Omega$ is called a domain of injective holomorphy if
there exists an injective holomorphic function
$f\colon \Omega\rightarrow\mathbb{C}$ that is non-extendable. We give examples of
domains that are domains of injective holomorphy and others that
are not. In particular, every regular domain
$(\overline\Omega^o=\Omega)$ is a domain of injective holomorphy, and
every simply connected domain is a domain of injective holomorphy
as well.
Keywords:domains of holomorphy Category:30Exx |
13. CMB 2009 (vol 53 pp. 23)
Boundedness From Below of Multiplication Operators Between $\alpha$-Bloch Spaces In this paper, the boundedness from below of multiplication
operators between $\alpha$-Bloch spaces $\mathcal B^\alpha$, $\alpha\gt 0$, on the
unit disk $D$ is studied completely. For a bounded multiplication
operator $M_u\colon \mathcal B^\alpha\to\mathcal B^\beta$, defined by $M_uf=uf$ for
$f\in\mathcal B^\alpha$, we prove the following result:
(i) If $0\lt \beta\lt \alpha$, or
$0\lt \alpha\le1$ and $\alpha\lt \beta$, $M_u$ is not bounded below;
(ii) if $0\lt \alpha=\beta\le1$, $M_u$ is bounded below if and only if
$\liminf_{z\to\partial D}|u(z)|\gt 0$;
(iii) if $1\lt \alpha\le\beta$, $M_u$ is
bounded below if and only if there exist a $\delta\gt 0$ and a positive
$r\lt 1$ such that for every point $z\in D$ there is a point $z'\in
D$ with the property $d(z',z)\lt r$ and
$(1-|z'|^2)^{\beta-\alpha}|u(z')|\ge\delta$, where $d(\cdot,\cdot)$ denotes
the pseudo-distance on $D$.
Keywords:$\alpha$-Bloch function, multiplication operator Categories:32A18, 30H05 |
14. CMB 2009 (vol 52 pp. 481)
Some Infinite Products of Ramanujan Type In his ``lost'' notebook, Ramanujan stated two results, which are equivalent to the identities
\[
\prod_{n=1}^{\infty} \frac{(1-q^n)^5}{(1-q^{5n})}
=1-5\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{d} d \Big) q^n
\]
and
\[
q\prod_{n=1}^{\infty} \frac{(1-q^{5n})^5}{(1-q^{n})}
=\sum_{n=1}^{\infty}\Big( \sum_{d \mid n} \qu{5}{n/d} d \Big) q^n.
\]
We give several more identities of this type.
Keywords:Power series expansions of certain infinite products Categories:11E25, 11F11, 11F27, 30B10 |
15. CMB 2009 (vol 52 pp. 53)
Cusp Forms Like $\Delta$ Let $f$ be a square-free integer and denote by $\Gamma_0(f)^+$ the
normalizer of $\Gamma_0(f)$ in $\SL(2,\R)$. We find the analogues of
the cusp form $\Delta$ for the groups $\Gamma_0(f)^+$.
Categories:11F03, 11F22, 30F35 |
16. CMB 2008 (vol 51 pp. 481)
Universal Inner Functions on the Ball It is shown that given any sequence of automorphisms $(\phi_k)_k$ of the
unit ball $\bn$ of $\cn$ such that $\|\phi_k(0)\|$ tends to $1$,
there exists an inner function
$I$ such that the family of ``non-Euclidean translates"
$(I\circ\phi_k)_k$ is locally uniformly dense in the unit ball of
$H^\infty(\bn)$.
Keywords:inner functions, automorphisms of the ball, universality Categories:32A35, 30D50, 47B38 |
17. CMB 2008 (vol 51 pp. 497)
Expected Norms of Zero-One Polynomials Let $\cA_n = \big\{ a_0 + a_1 z + \cdots + a_{n-1}z^{n-1} : a_j \in \{0, 1 \
} \big\}$, whose elements are called \emf{zero-one polynomials}
and correspond naturally to the $2^n$ subsets of $[n] := \{ 0, 1,
\ldots, n-1 \}$. We also let $\cA_{n,m} = \{ \alf(z) \in \cA_n :
\alf(1) = m \}$, whose elements correspond to the ${n \choose m}$
subsets of~$[n]$ of size~$m$, and let $\cB_n = \cA_{n+1} \setminus
\cA_n$, whose elements are the zero-one polynomials of degree
exactly~$n$.
Many researchers have studied norms of polynomials with restricted
coefficients. Using $\norm{\alf}_p$ to denote the usual $L_p$ norm
of~$\alf$ on the unit circle, one easily sees that $\alf(z) = a_0 +
a_1 z + \cdots + a_N z^N \in \bR[z]$ satisfies $\norm{\alf}_2^2 = c_0$
and $\norm{\alf}_4^4 = c_0^2 + 2(c_1^2 + \cdots + c_N^2)$, where $c_k
:= \sum_{j=0}^{N-k} a_j a_{j+k}$ for $0 \le k \le N$.
If $\alf(z) \in \cA_{n,m}$, say $\alf(z) = z^{\beta_1} + \cdots +
z^{\beta_m}$ where $\beta_1 < \cdots < \beta_m$, then $c_k$ is the
number of times $k$ appears as a difference $\beta_i - \beta_j$. The
condition that $\alf \in \cA_{n,m}$ satisfies $c_k \in \{0,1\}$ for $1
\le k \le n-1$ is thus equivalent to the condition that $\{ \beta_1,
\ldots, \beta_m \}$ is a \emf{Sidon set} (meaning all differences of
pairs of elements are distinct).
In this paper, we find the average of~$\|\alf\|_4^4$ over $\alf \in
\cA_n$, $\alf \in \cB_n$, and $\alf \in \cA_{n,m}$. We further show
that our expression for the average of~$\|\alf\|_4^4$ over~$\cA_{n,m}$
yields a new proof of the known result: if $m = o(n^{1/4})$ and
$B(n,m)$ denotes the number of Sidon sets of size~$m$ in~$[n]$, then
almost all subsets of~$[n]$ of size~$m$ are Sidon, in the sense that
$\lim_{n \to \infty} B(n,m)/\binom{n}{m} = 1$.
Categories:11B83, 11C08, 30C10 |
18. CMB 2008 (vol 51 pp. 334)
Value Distribution of the Riemann Zeta Function In this note, we give a new short proof of the fact, recently
discovered by Ye, that all (finite) values are equidistributed by
the Riemann zeta function.
Keywords:Nevanlinna theory, deficiency, Riemann zeta function Category:30D35 |
19. CMB 2008 (vol 51 pp. 195)
Boundedness from Below of Composition Operators on $\alpha$-Bloch Spaces We give a necessary and sufficient condition for a composition
operator on an $\alpha$-Bloch space with $\alpha\ge 1$ to be bounded below.
This extends a known result for the Bloch space due to P. Ghatage,
J. Yan, D. Zheng, and H. Chen.
Keywords:Bloch functions, composition operators Categories:32A18, 30H05 |
20. CMB 2007 (vol 50 pp. 579)
$p$-Radial Exceptional Sets and Conformal Mappings For $p>0$ and for a given set $E$ of type $G_{\delta}$ in the boundary
of the unit disc $\partial\mathbb D$ we construct a holomorphic function
$f\in\mathbb O(\mathbb D)$ such that
\[
\int_{\mathbb D\setminus[0,1]E}|ft|^{p}\,d\mathfrak{L}^{2}<\infty\]
and\[
E=E^{p}(f)=\Bigl\{ z\in\partial\mathbb D:\int_{0}^{1}|f(tz)|^{p}\,dt=\infty\Bigr\} .\]
In particular if a set $E$ has a measure equal to zero, then a function
$f$ is constructed as integrable with power $p$ on the unit disc $\mathbb D$.
Keywords:boundary behaviour of holomorphic functions, exceptional sets Categories:30B30, 30E25 |
21. CMB 2007 (vol 50 pp. 11)
van der Pol Expansions of L-Series We provide concise series representations for various
L-series integrals. Different techniques are needed below and above
the abscissa of absolute convergence of the underlying L-series.
Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, L-series Categories:11M35, 11M41, 30B50 |
22. CMB 2007 (vol 50 pp. 123)
Simultaneous Approximation and Interpolation on Arakelian Sets We extend results of P.~M. Gauthier, W. Hengartner and
A.~A. Nersesyan
on simultaneous approximation and interpolation
on Arakelian sets.
Keywords:Arakelian's theorem,, Arakelian sets Category:30E10 |
23. CMB 2006 (vol 49 pp. 381)
On the Membership in Bergman Spaces of the Derivative of a Blaschke Product With Zeros in a Stolz Domain |
On the Membership in Bergman Spaces of the Derivative of a Blaschke Product With Zeros in a Stolz Domain It is known that the derivative of
a Blaschke product whose zero sequence lies in a Stolz angle
belongs to all the Bergman spaces $A^p$ with $0
1$). As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that $B'\notin A^{3/2}$. Keywords:Blaschke products, Hardy spaces, Bergman spaces Categories:30D50, 30D55, 32A36 |
24. CMB 2006 (vol 49 pp. 438)
Unimodular Roots of\\ Special Littlewood Polynomials We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood
polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal
if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if
$n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ for all $j$. It has been observed
that Littlewood polynomials with particularly high minimum modulus on
the unit
circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a
skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle,
as well as providing a new proof of the known result that a self-reciprocal
Littlewood polynomial must have a zero on the unit circle.
Categories:26C10, 30C15, 42A05 |
25. CMB 2005 (vol 48 pp. 580)
Exceptional Sets in Hartogs Domains Assume that $\Omega$ is a Hartogs domain in $\mathbb{C}^{1+n}$,
defined as $\Omega=\{(z,w)\in\mathbb{C}^{1+n}:|z|<\mu(w),w\in H\}$, where $H$ is an open set in
$\mathbb{C}^{n}$ and $\mu$ is a continuous function with positive values in $H$ such that $-\ln\mu$
is a strongly plurisubharmonic function in $H$. Let $\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$.
For a given set $E$ contained in $H$ of the type $G_{\delta}$ we construct a holomorphic function
$f\in\mathbb{O}(\Omega)$ such that
\[
E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}|f(\cdot\,,w)|^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}.
\]
Keywords:boundary behaviour of holomorphic functions,, exceptional sets Category:30B30 |