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26. CMB 2011 (vol 56 pp. 194)

Stefánsson, Úlfar F.
 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 polynomialsCategories:42C05, 42C99, 41A60, 30B50

27. CMB 2011 (vol 55 pp. 509)

Gauthier, P. M.; Nestoridis, V.
 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 holomorphyCategory:30Exx

28. 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 operatorCategories:32A18, 30H05

29. 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 productsCategories:11E25, 11F11, 11F27, 30B10

30. 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

31. 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, universalityCategories:32A35, 30D50, 47B38

32. 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

33. 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 functionCategory:30D35

34. 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 operatorsCategories:32A18, 30H05

35. 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 setsCategories:30B30, 30E25

36. 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-seriesCategories:11M35, 11M41, 30B50

37. 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 setsCategory:30E10

38. 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

39. 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 spacesCategories:30D50, 30D55, 32A36

40. 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 setsCategory:30B30

41. 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 functionsCategories:32A35, 30D50, 47B38

42. 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

43. 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 valueCategory:30D35

44. 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

45. 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 functionCategory:30

46. 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, FrÃ©chet envelopeCategories:46E10, 30A78, 30A76

47. 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

48. 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 surfacesCategories:30C62, 31A05, 31A20, 49Q05

49. 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

50. 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
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