Approximation by walsh-kaczmarz-fejér means on the hardy space

The main aim of this paper is to find necessary and sufficient conditions for the convergence of Walsh-Kaczmarz-Fejér means in the terms of the modulus of continuity on the Hardy spaces H_p, when 0<p≤ 1/2.     

A limit theorem for Szegö polynomials with respect to convolution of point masses with the Fejér kernel

Two recently-proposed methods for estimating the m frequencies of a trigonometric signal using Szegö polynomials of fixed degree k>m consist of multiplying the moments of the n-truncated periodogram by the moments of the Poisson kernel and the wrapped Gaussian, respectively, in an effort to address the non-convergence of the polynomials as n→∞. These methods are seen to be equivalent to convolution of point masses with approximate identities, suggesting a general method. We characterize the limit polynomial for the case when the approximate identity is the Fejér kernel, extending recent results of the author for the case of the Poisson kernel. Moreover, the limit is seen to be the same as in the former case.     

Réflexivité d'une extension d'un opérateur sous-normal par un opérateur algébrique

A bounded linear operator on a Hilbert space is said to be reflexive if the operators which leave invariant the invariant subspaces of T are wot-limits of polynomials in T. In this paper we give a necessary and sufficient condition for an extension of a subnormal operator by an algebraic one to be reflexive.We also give a formula for the reflexivity defect of such extensions.     

Sur l'existence, l'unicité, la stabilité et les propriétés de grandes déviations des solutions d'équations différentielles stochastiques rétrogrades à horizon aléatoire. Application à des problèmes de perturbations singulières

We study the existence, uniqueness and stability of solutions of backward stochastic differential equations with random terminal time under new assumptions; then we establish a large deviation principle for the solutions of such equations, related to a family of Markov processes, the diffusion coefficient of which tends to zero. Finally we apply these results to the analysis of some singular perturbation problems for a class of nonlinear partial differential equations.     

Réflexivité d’une extension d’un opérateur normal par un opérateur nilpotent

Let T be an extension of a one to one normal operator A by a nilpotent operator N. In this paper we calculate the defect of reflexivity of T. We give a necessary and sufficient condition to insure the reflexivity of such extensions. In particular, it is shown that T is reflexive when N is reflexive.     

Quelques aspects sur la géométrie des surfaces algébriques réelles

Let f be a real polynomial of degree n?3 in two variables. It is known that its hessian is a real polynomial in two variables of degree at most 2n−4. In 1876, A. Harnack prove that the number of connected components of an algebraic plane curve of degree m embedded in is at most (m−1)(m−2)/2+1. So, by A. Harnack, the number of compact connected components of the parabolic curve of the graph of f is at most (2n−5)(n−3)+1. Until now, we do not know if this bound is optimal for n?4.In this note we give a class of real polynomials of degree n?3 in two variables such that the parabolic curve of the graph of each polynomial being to this class has exactly (n−1)(n−2)/2 connected components and exactly n(n−2) special parabolic points. Moreover, all these parabolic curves are smooth and compact.It is known that for each smooth algebraic surface of degree n?3 embedded in , the maximal number of connected components of its parabolic curve is 2n(n−2)(5n−12)+2. Until now, we do not know if this bound is optimal. In this note we give a family of smooth algebraic surfaces of degree n?3 embedded in . The parabolic curve of each surface being to this family is smooth and it has at least n(n−1)/(n−2)2 connected components and at least n²(n−2) special parabolic points.     

Series expansions in Fréchet spaces and their duals, construction of Fréchet frames

Frames for Fréchet spaces X_F with respect to Fréchet sequence spaces Θ_F are studied, and conditions implying series expansions in X_F and are determined. If is a Θ₀-frame for X₀ and Θ_F (resp. X_F) is given, we construct a sequence {X_s}_s∈N0, X_s⊂X_s−1, s∈N, (resp. {Θ_s}_s∈N0, Θ_s⊂Θ_s−1, s∈N), so that is a pre-F-frame or F-frame for X_F with respect to Θ_F under different assumptions given on X₀, Θ₀ and Θ_F (resp. X_F).     

Généralisation des inégalités ergodiques maximales

Résumé On établit des inégalités maximales du type de Hopf et de Brunel dans un cadre très général. On en déduit une extension du théorème ergodique de Chacon-Ornstein qui unifie plusieurs résultats classiques et en fournit de nouveaux.     

On the number of (r,r+1)‐ factors in an (r,r+1)‐factorization of a simple graph

For integers d≥0, s≥0, a (d, d+s)‐graph is a graph in which the degrees of all the vertices lie in the set {d, d+1, …, d+s}. For an integer r≥0, an (r, r+1)‐factor of a graph G is a spanning (r, r+1)‐subgraph of G. An (r, r+1)‐factorization of a graph G is the expression of G as the edge‐disjoint union of (r, r+1)‐factors. For integers r, s≥0, t≥1, let f(r, s, t) be the smallest integer such that, for each integer d≥f(r, s, t), each simple (d, d+s) ‐graph has an (r, r+1) ‐factorization with x (r, r+1) ‐factors for at least t different values of x. In this note we evaluate f(r, s, t). © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 257‐268, 2009     

Equations différentielles invariantes sur les groupes et algèbres de Lie réductifs II

Dans un article précédent, nous avons démontré que si D est un opérateur différentiel bi-invariant sur un groupe réductif G vérifiant la condition de Benabdallah-Rouvière, alors on peut résoudre l’équation différentielle Du=v dans l'espace des distributions G-invariantes (par automorphismes intérieurs) d'ordre fini; nous allons montrer ici que, sous la même hypothèse, on peut résoudre cette équation dans l'espace de toutes les distributions G-invariantes. D'autre part, nous donnons un exemple dans qui montre que les équations différentielles invariantes dans les algèbres de Lie réductives ne sont pas toujours résolubles dans l'espace des fonctions indéfiniment différentiables invariantes.     

Sharp Békollé estimates for the Bergman projection

We prove sharp estimates for the Bergman projection in weighted Bergman spaces in terms of the Békollé constant. Our main tools are a dyadic model dominating the operator and an adaptation of a method of Cruz-Uribe, Martell and Pérez.     

Sur la caractéristique d'Euler des feuilletages mesurés

On prouve que la caractéristique d'Euler d'un feuilletage mesuré est nulle si et seulement si celui-ci admet un champ tangent transversalement mesurable dont l'ensemble de singularités est de mesure arbitrairement petite. Si le feuilletage est moyennable, alors on peut construire un tel champ sans singularités. Si les feuilles sont de dimension deux, l'annulation de la caractéristique d'Euler implique la moyennabilité ; plus encore, elle équivaut à l'existence d'une action mesurable de R² sur la variété ambiant qui est continue et localement libre sur chaque feuille.     

Valeurs d'une fonction de Picard en des points algébriques

Using geometric tools introduced by P. Cohen, H. Shiga, J. Wolfart and G. Wüstholz, we show in Theorem 1 that when a certain Gauss hypergeometric function takes an algebraic value at an algebraic point, then another Gauss hypergeometric function takes a transcendental value at a related algebraic point. Using Appell hypergeometric functions, which generalize to two variables the Gauss functions, we study values at algebraic points of a new transcendental function defined in terms of these two functions. By Theorem 2, these values correspond to abelian varieties in the same isogeny class. Using a result of Edixhoven-Yafaev [B. Edixhoven, A. Yafaev, Subvarieties of Shimura varieties, Ann. of Math. 157 (2003) 621-645], this last result is in turn related to the distribution of the moduli of such abelian varieties in certain Shimura varieties.     

A direct approach to convergence of multivariate,nonhomogeneous, Padé approximants

We present a direct approach for proving convergence in measure/product capacity of multivariate, nonhomogeneous, Padé approximants. Previous approaches have involved projection onto Padé-type approximation in one variable, and only yielded convergence in (Lebesgue) measure.     

Universal characters, integrable chains and the Painlevé equations

The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see (Adv. Math. 74 (1989) 57). We prove that the universal character solves the Darboux chain. The N-periodic closing of the chain is equivalent to the Painlevé equation of type . Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters.     

Multivariate Interpolation Using Carlitz's Inversion Formulas

Multivariate rational exponential Lagrange interpolation formulas, Hermite interpolation formulas, and Hermite–Fejér interpolation formulas of the Newton type are established by using Carlitz's inversion formulas. The recurrence relation for constructing Lagrange interpolation is also given. In addition, by setting q1 in the obtained formulas, we obtain the corresponding polynomial interpolation formulas with combinatorial form.     

On tronquée solutions of the first Painlevé hierarchy

It is well known that, due to Boutroux, the first Painlevé equation admits solutions characterized by divergent asymptotic expansions near infinity in specified sectors of the complex plane. In this paper, we show that such solutions exist for higher order analogues of the first Painlevé equation (the first Painlevé hierarchy) as well.