Augustin Mouze

ON THE FREQUENT UNIVERSALITY OF UNIVERSAL TAYLOR SERIES IN THE COMPLEX PLANE

We prove that the classical universal Taylor series in the complex plane are never frequently universal. On the other hand, we prove the 1-upper frequent universality of all these universal Taylor series.

Universality and ultradifferentiable functions: Fekete’s theorem

The purpose of this article is to establish extensions of Feketes Theorem concerning the existence of universal power series of functions defined by estimates on successive derivatives.

Determination of a Universal Series

The known proofs for universal Taylor series do not determine a specific universal Taylor series. In the present paper, we isolate a specific universal Taylor series by modifying the proof in [30]. Thus we determine all Taylor coefficients of a specific universal Taylor series on the disc or on a polygonal domain. Furthermore in non simply connected domains, when universal Taylor series exist, we can construct a sequence of specific rational functions converging to a universal function, provided the boundary is good enough. The solution uses an infinite denumerable procedure and a finite number of steps is not sufficient. However we solve a Runge’s type problem in a finite number of steps.

Universality and summability of Dirichlet series

We show genericity and algebraic genericity of Dirichlet series, whose given matrix transform has universal properties. Conversely, for any given formal Dirichlet series L with non-zero coefficients, we construct a matrix A, so that the A-transform of L is universal. This extends results of Bernal-González, Calderón-Moreno and Luh. We also show that uniform approximation on admissible compact sets with empty interior implies approximation in the sense of Menchoff with respect to any σ-finite Borel measure. Finally, we prove that universal Dirichlet series in the sense of Bayart are not (R, 1) summable on the boundary iℝ, where (R, 1) is the summability method of logarithmic mean type.

Universal approximation theorem for Dirichlet series

The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right half of the complex plane. The derivation operator used in the analytic case is substituted by a weighted backward shift operator in the Dirichlet case. We show the similarities and extensions in comparing both results. Several density results are proved that finally lead to the main theorem on simultaneous approximation.

On doubly universal functions

Abstract Let ( λ n ) be a strictly increasing sequence of positive integers. Inspired by the notions of topological multiple recurrence and disjointness in dynamical systems, Costakis and Tsirivas have recently established that there exist power series ∑ k ≥ 0 a k z k with radius of convergence 1 such that the pairs of partial sums { ( ∑ k = 0 n a k z k , ∑ k = 0 λ n a k z k ) : n = 1 , 2 , … } approximate all pairs of polynomials uniformly on compact subsets K ⊂ { z ∈ C : | z | ≥ 1 } , with connected complement, if and only if lim sup n λ n n = + ∞ . In the present paper, we give a new proof of this statement avoiding the use of tools of potential theory. It allows to study the case of doubly universal infinitely differentiable functions. We obtain also the algebraic genericity of the set of such power series. Further we show that the Cesaro means of partial sums of power series with radius of convergence 1 cannot be frequently universal.

Un théorème d'Artin pour des anneaux de séries formelles à croissance contrôlée

Resume Soit f(X,Y) = (ƒ1(X,Y),…,ƒm(X,Y)) = 0 un systeme dequations, ou les fi, i = 1,…, m, sont des series formelles, en X = (X1,…, Xd) et Y = (Y1,…, Yq), dont les coefficients satisfont des conditions de croissance. On suppose que ce systeme admet une solution formelle y(X), (0) = 0, telle que f(X,y (X)) = 0. On prouve alors quil existe y(X, t) = (y1 (X, t),…, yq (X, t)), des series formelles qui satisfont les memes conditions de croissance, ou t = (t1,…,ts) sont des nouvelles variables, et t(X) = (t1(X),…, ts(X)), des series formelles sans terme constant, telles que y(X) = y(X,t(X)) et ƒ(X,y(X,t)) = 0.

A quantitative interpretation of the frequent hypercyclicity criterion

We give a quantitative interpretation of the Frequent Hypercyclicity Criterion. Actually we show that an operator which satisfies the Frequent Hypercyclicity Criterion is necessarily A-frequently hypercyclic, where A refers to some weighted densities sharper than the natural lower density. In that order, we exhibit different scales of weighted densities that are of interest to quantify the frequency measured by the Frequent Hypercyclicity Criterion. Moreover we construct an example of unilateral weighted shift which is frequently hypercyclic but not A-frequently hypercyclic on a particular scale.

Sur l'irréductibilité dans l'anneau des séries de Dirichlet analytiques

We discuss some local analytic properties of the ring of Dirichlet series. We obtain mainly the equivalence between the irreducibility in the analytic ring and in the formal one. In the same way we prove that the ring of analytic Dirichlet series is integrally closed in the ring of formal Dirichlet series. Finally we introduce the notion of standard basis in these rings and we give a finitely generated ideal which does not admit standard bases.

Factorialité de l'anneau des séries de Dirichlet analytiques

Resume On etudie des proprietes arithmetiques de lanneau des series de Dirichlet analytiques. En particulier, on prouve sa factorialite, en obtenant un resultat de division par plusieurs series. Pour citer cet article : F.xa0Bayart, A.xa0Mouze, C. R. Acad. Sci. Paris, Ser. I 336 (2003).