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26. CMB 2009 (vol 53 pp. 23)

Chen, Huaihui; Zhang, Minzhu
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

27. CMB 2009 (vol 52 pp. 481)

Alaca, Ay\c{s}e; Alaca, \c{S}aban; Williams, Kenneth S.
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

28. CMB 2009 (vol 52 pp. 53)

Cummins, C. J.
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

29. CMB 2008 (vol 51 pp. 497)

Borwein, Peter; Choi, Kwok-Kwong Stephen; Mercer, Idris
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

30. CMB 2008 (vol 51 pp. 481)

Bayart, Frédéric
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

31. CMB 2008 (vol 51 pp. 334)

Ascah-Coallier, I.; Gauthier, P. M.
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

32. CMB 2008 (vol 51 pp. 195)

Chen, Huaihui; Gauthier, Paul
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

33. CMB 2007 (vol 50 pp. 579)

Kot, Piotr
$p$-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

34. CMB 2007 (vol 50 pp. 123)

Nikolov, Nikolai; Pflug, Peter
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

35. CMB 2007 (vol 50 pp. 11)

Borwein, David; Borwein, Jonathan
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

36. CMB 2006 (vol 49 pp. 381)

Girela, Daniel; Peláez, José Ángel
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 $01$). 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

37. CMB 2006 (vol 49 pp. 438)

Mercer, Idris David
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

38. CMB 2005 (vol 48 pp. 580)

Kot, Piotr
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

39. CMB 2005 (vol 48 pp. 409)

Gauthier, P. M.; Xiao, J.
The Existence of Universal Inner Functions on the Unit Ball of $\mathbb{C}^n$
It is shown that there exists an inner function $I$ defined on the unit ball ${\bf B}^n$ of ${\mathbb C}^n$ such that each function holomorphic on ${\bf B}^n$ and bounded by $1$ can be approximated by ``non-Euclidean translates" of $I$.

Keywords:universal inner functions
Categories:32A35, 30D50, 47B38

40. CMB 2004 (vol 47 pp. 17)

Gorkin, Pamela; Mortini, Raymond
Universal Singular Inner Functions
We show that there exists a singular inner function $S$ which is universal for noneuclidean translates; that is one for which the set $\{S(\frac{z+z_n}{1+\bar z_nz}):n\in\mathbb{N}\}$ is locally uniformly dense in the set of all zero-free holomorphic functions in $\mathbb{D}$ bounded by one.

Category:30D50

41. CMB 2004 (vol 47 pp. 152)

Zheng, Jian-Hua
On Uniqueness of Meromorphic Functions with Shared Values in Some Angular Domains
In this paper we investigate the uniqueness of transcendental meromorphic function dealing with the shared values in some angular domains instead of the whole complex plane.

Keywords:Nevanlinna theory, meromorphic function, shared value
Category:30D35

42. CMB 2003 (vol 46 pp. 559)

Marco, Nicolas; Massaneda, Xavier
On Density Conditions for Interpolation in the Ball
In this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of $\C^n$, $n>1$. We first give density conditions for a sequence to be interpolating for the class $A^{-\infty}$ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension $1$, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for $n=1$ coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space.

Categories:32A36, 32A38, 30E05

43. CMB 2003 (vol 46 pp. 95)

Gauthier, P. M.
Cercles de remplissage for the Riemann Zeta Function
The celebrated theorem of Picard asserts that each non-constant entire function assumes every value infinitely often, with at most one exception. The Riemann zeta function has this Picard behaviour in a sequence of discs lying in the critical band and whose diameters tend to zero. According to the Riemann hypothesis, the value zero would be this (unique) exceptional value.

Keywords:cercles de remplissage, Riemann zeta function
Category:30

44. CMB 2002 (vol 45 pp. 265)

Nawrocki, Marek
On the Smirnov Class Defined by the Maximal Function
H.~O.~Kim has shown that contrary to the case of $H^p$-space, the Smirnov class $M$ defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, {\it i.e.} they have the same dual spaces and the same Fr\'echet envelopes. We describe a general form of a continuous linear functional on $M$ and multiplier from $M$ into $H^p$, $0 < p \leq \infty$.

Keywords:Smirnov class, maximal radial function, multipliers, dual space, Fréchet envelope
Categories:46E10, 30A78, 30A76

45. CMB 2002 (vol 45 pp. 89)

Grant, David
On Gunning's Prime Form in Genus $2$
Using a classical generalization of Jacobi's derivative formula, we give an explicit expression for Gunning's prime form in genus 2 in terms of theta functions and their derivatives.

Categories:14K25, 30F10

46. CMB 2002 (vol 45 pp. 154)

Weitsman, Allen
On the Poisson Integral of Step Functions and Minimal Surfaces
Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

Keywords:harmonic mappings, dilatation, minimal surfaces
Categories:30C62, 31A05, 31A20, 49Q05

47. CMB 2002 (vol 45 pp. 36)

Cummins, C. J.
Modular Equations and Discrete, Genus-Zero Subgroups of $\SL(2,\mathbb{R})$ Containing $\Gamma(N)$
Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains $\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there is a unique normalised generator of the field of $G$-automorphic functions which is known as a normalised Hauptmodul. This paper gives a characterisation of normalised Hauptmoduls as formal $q$ series using modular polynomials.

Categories:11F03, 11F22, 30F35

48. CMB 2001 (vol 44 pp. 420)

Gauthier, P. M.; Pouryayevali, M. R.
Approximation by Meromorphic Functions With Mittag-Leffler Type Constraints
Functions defined on closed sets are simultaneously approximated and interpolated by meromorphic functions with prescribed poles and zeros outside the set of approximation.

Categories:30D30, 30E10, 30E15

49. CMB 2000 (vol 43 pp. 183)

Ionesei, Gheorghe
A Gauge Theoretic Proof of the Abel-Jacobi Theorem
We present a new, simple proof of the classical Abel-Jacobi theorem using some elementary gauge theoretic arguments.

Keywords:Abel-Jacobi theorem, abelian gauge theory
Categories:58D27, 30F99

50. CMB 2000 (vol 43 pp. 115)

Schmutz Schaller, Paul
Perfect Non-Extremal Riemann Surfaces
An infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group $\PSL(2,{\sf Z})$ of level $6$. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function {\it syst\/} which associates to every Riemann surface $M$ the length of a systole, which is a shortest closed geodesic of $M$.

Categories:11H99, 11F06, 30F45
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