Certain problems concerning polynomials and transcendental entire functions of exponential type

Soit $\displaystyle P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ un polynôme de degré $n$ et $\displaystyle M:=\sup_{|z|=1}|P(z)|.$ Sans aucne restriction suplémentaire, on sait que $|P'(z)|\leq Mn$ pour $|z|\leq 1$ (inégalité de Bernstein). Si nous supposons maintenant que les zéros du polynôme $P$ sont à l'extérieur du cercle $|z|=k,$ quelle amélioration peut-on apporter à l'inégalité de Bernstein? Il est déjà connu [{\bf \ref{Mal1}}] que dans le cas où $k\geq 1$ on a $$(*) \qquad |P'(z)|\leq \frac{n}{1+k}M \qquad (|z|\leq 1),$$ qu'en est-il pour le cas où $k < 1$? Quelle est l'inégalité analogue à $(*)$ pour une fonction entière de type exponentiel $\tau ?$ D'autre part, si on suppose que $P$ a tous ses zéros dans $|z|\geq k \, \, (k\geq 1),$ quelle est l'estimation de $|P'(z)|$ sur le cercle unité, en terme des quatre premiers termes de son développement en série entière autour de l'origine. Cette thèse constitue une contribution à la théorie analytique des polynômes à la lumière de ces questions.

Vector spaces of entire functions of unbounded type

Let E be an infinite dimensional complex Banach space. We prove the existence of an infinitely generated algebra, an infinite dimensional closed subspace and a dense subspace of entire functions on E whose non-zero elements are functions of unbounded type. We also show that the τδ topology on the space of all holomorphic functions cannot be obtained as a countable inductive limit of Fr´echet spaces. RESUMEN. Sea E un espacio de Banach complejo de dimensión infinita y sea H(E) el espacio de funciones holomorfas definidas en E. En el artículo se demuestra la existencia de un álgebra infinitamente generada en H(E), un subespacio vectorial en H(E) cerrado de dimensión infinita y un subespacio denso en H(E) cuyos elementos no nulos son funciones de tipo no acotado. También se demuestra que el espacio de funciones holomorfas con la topología ? no es un límite inductivo numberable de espacios de Fréchet.

Geometrical logarithmic and trigonometric hypercomplex functions of quaternionic type

Quaternionic theory has greatly been developed in recent years [1-12]. Thus, in our view, the study of trigonometric and logarithmic type quaternionic functions is important for the determination and realization of a hyper complex theory. In this paper, we intend to give a geometrical foundation for both logarithmic and trigonometric hyper complex functions based on the exponential function of quaternionic type recently introduced by Borges, Marão and Machado in their paper entitled Geometrical octonions II: Hyper regularity and hyper periodicity of the exponential function appearing. © 2011 Pushpa Publishing House.

Generalized Sobolev Spaces of Exponential Type Associated with the Dunkl Operators

2000 Mathematics Subject Classification: 35E45

On a Ratio of Functions of Exponential Random Variables and Some Applications

Consider L independent and identically distributed exponential random variables (r.vs) X-1, X-2 ,..., X-L and positive scalars b(1), b(2) ,..., b(L). In this letter, we present the probability density function (pdf), cumulative distribution function and the Laplace transform of the pdf of the composite r.v Z = (Sigma(L)(j=1) X-j)(2) / (Sigma(L)(j=1) b(j)X(j)). We show that the r.v Z appears in various communication systems such as i) maximal ratio combining of signals received over multiple channels with mismatched noise variances, ii)M-ary phase-shift keying with spatial diversity and imperfect channel estimation, and iii) coded multi-carrier code-division multiple access reception affected by an unknown narrow-band interference, and the statistics of the r.v Z derived here enable us to carry out the performance analysis of such systems in closed-form.

Formules de quadrature pour les fonctions entières de type exponentiel

Ce mémoire contient quelques résultats sur l'intégration numérique. Ils sont liés à la célèbre formule de quadrature de K. F. Gauss. Une généralisation très intéressante de la formule de Gauss a été obtenue par P. Turán. Elle est contenue dans son article publié en 1948, seulement quelques années après la seconde guerre mondiale. Étant données les circonstances défavorables dans lesquelles il se trouvait à l'époque, l'auteur (Turán) a laissé beaucoup de détails à remplir par le lecteur. Par ailleurs, l'article de Turán a inspiré une multitude de recherches; sa formule a été étendue de di érentes manières et plusieurs articles ont été publiés sur ce sujet. Toutefois, il n'existe aucun livre ni article qui contiennent un compte-rendu détaillé des résultats de base, relatifs à la formule de Turán. Je voudrais donc que mon mémoire comporte su samment de détails qui puissent éclairer le lecteur tout en présentant un exposé de ce qui a été fait sur ce sujet. Voici comment nous avons organisé le contenu de ce mémoire. 1-a. La formule de Gauss originale pour les polynômes - L'énoncé ainsi qu'une preuve. 1-b. Le point de vue de Turán - Compte-rendu détaillé des résultats de son article. 2-a. Une formule pour les polynômes trigonométriques analogue à celle de Gauss. 2-b. Une formule pour les polynômes trigonométriques analogue à celle de Turán. 3-a. Deux formules pour les fonctions entières de type exponentiel, analogues à celle de Gauss pour les polynômes. 3-b. Une formule pour les fonctions entières de type exponentiel, analogue à celle de Turán. 4-a. Annexe A - Notions de base sur les polynômes de Legendre. 4-b. Annexe B - Interpolation polynomiale. 4-c. Annexe C - Notions de base sur les fonctions entières de type exponentiel. 4-d. Annexe D - L'article de P. Turán.

On real analytic functions of unbounded type

In this paper we prove several results on the existence of analytic functions on an infinite dimensional real Banach space which are bounded on some given collection of open sets and unbounded on others. In addition, we also obtain results on the density of some subsets of the space of all analytic functions for natural locally convex topologies on this space. RESUMEN. Los autores demuestran varios resultados de existencia de funciones analíticas en espacios de Banach reales de dimensión infinita que están acotadas en un colección de subconjuntos abiertos y no acotadas en los conjuntos de otra colección. Además, se demuestra la densidad de ciertos subconjuntos de funciones analíticas para varias topologías localmente convexas.

Exponential approximation in the norms and semi-norms

The deviations of some entire functions of exponential type from real-valued functions and their derivatives are estimated. As approximation metrics we use the Lp-norms and power variations on R. Theorems presented here correspond to the Ganelius and Popov results concerning the one-sided trigonometric approximation of periodic functions (see [4, 5 and 8]). Some related facts were announced in [2, 3, 6 and 7].

Sur les comportements locaux de polynômes et polynômes trigonométriques

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Disjoint mixing operators

Chan and Shapiro showed that each (non-trivial) translation operator acting on the Fréchet space of entire functions endowed with the topology of locally uniform convergence supports a universal function of exponential type zero. We show the existence of d-universal functions of exponential type zero for arbitrary finite tuples of pairwise distinct translation operators. We also show that every separable infinite-dimensional Fréchet space supports an arbitrarily large finite and commuting disjoint mixing collection of operators. When this space is a Banach space, it supports an arbitrarily large finite disjoint mixing collection of C0-semigroups. We also provide an easy proof of the result of Salas that every infinite-dimensional Banach space supports arbitrarily large tuples of dual d-hypercyclic operators, and construct an example of a mixing Hilbert space operator T so that (T,T2) is not d-mixing.

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

We consider a modification of the three-dimensional Navier-Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e(vertical bar k vertical bar/kd) at high wavenumbers vertical bar k vertical bar. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e-(C(k/kd) ln(vertical bar k vertical bar/kd)) for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C-* = 1/ ln 2. The same behavior with a universal constant C-* is conjectured for the Navier-Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier-Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.

Novel Functions of (p)ppGpp and Cyclic di-GMP in Mycobacterial Physiology Revealed by Phenotype Microarray Analysis of Wild-Type and Isogenic Strains of Mycobacterium smegmatis

The bacterial second messengers (p)ppGpp and bis-(3'-5')-cyclic dimeric GMP (c-di-GMP) regulate important functions, such as transcription, virulence, biofilm formation, and quorum sensing. In mycobacteria, they regulate long-term survival during starvation, pathogenicity, and dormancy. Recently, a Pseudomonas aeruginosa strain lacking (p) ppGpp was shown to be sensitive to multiple classes of antibiotics and defective in biofilm formation. We were interested to find out whether Mycobacterium smegmatis strains lacking the gene for either (p)ppGpp synthesis (Delta rel(Msm)) or c-di-GMP synthesis (Delta dcpA) would display similar phenotypes. We used phenotype microarray technology to compare the growth of the wild-type and the knockout strains in the presence of several antibiotics. Surprisingly, the Delta rel(Msm) and Delta dcpA strains showed enhanced survival in the presence of many antibiotics, but they were defective in biofilm formation. These strains also displayed altered surface properties, like impaired sliding motility, rough colony morphology, and increased aggregation in liquid cultures. Biofilm formation and surface properties are associated with the presence of glycopeptidolipids (GPLs) in the cell walls of M. smegmatis. Thin-layer chromatography analysis of various cell wall fractions revealed that the levels of GPLs and polar lipids were reduced in the knockout strains. As a result, the cell walls of the knockout strains were significantly more hydrophobic than those of the wild type and the complemented strains. We hypothesize that reduced levels of GPLs and polar lipids may contribute to the antibiotic resistance shown by the knockout strains. Altogether, our data suggest that (p)ppGpp and c-di-GMP may be involved in the metabolism of glycopeptidolipids and polar lipids in M. smegmatis.

Maximum curves and isolated points of entire functions

Given M(r; f) =maxjzj=r (jf(z)j) , curves belonging to the set of points M = fz : jf(z)j = M(jzj; f)g were de�ned by Hardy to be maximum curves. Clunie asked the question as to whether the set M could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions f1(z) and f2(z), if the maximum curve of f1(z) is the real axis, conditions are found so that the real axis is also a maximum curve for the product function f1(z)f2(z). By means of these results an entire function of in�nite order is constructed for which the set M has an in�nite number of isolated points. A polynomial is also constructed with an isolated point.

Boundary value problems for analytic functions in the class of Cauchy-type integrals with density in

We study the Riemann boundary value problem , for analytic functions in the class of analytic functions represented by the Cauchy-type integrals with density in the spaces with variable exponent. We consider both the case when the coefficient is piecewise continuous and it may be of a more general nature, admitting its oscillation. The explicit formulas for solutions in the variable exponent setting are given. The related singular integral equations in the same setting are also investigated. As an application there is derived some extension of the Szegö-Helson theorem to the case of variable exponents.