We characterize the power series ${\sum }_{n=0}^{\infty }{c}_n{z}^n$ with the geometric property that, for sufficiently many points $z$, $\left|z\right|=1$, a circle $C\left(z\right)$ contains infinitely many partial sums. We show that ${\sum }_{n=0}^{\infty }{c}_n{z}^n$ is a rational function of special type; more precisely, there are $t\in ℝ$ and $n_0$, such that, the sequence $c_n{e}^{\mathrm{int}}$, $n\ge n_0$, is periodic. This result answers in the affirmative a question of J.-P. Kahane and furnishes stronger versions of the main results of [Katsoprinakis, Arkiv for Matematik]. We are led to consider special families of circles $C\left(z\right)$ with...

We give another version of the recently developed abstract theory of universal series to exhibit a necessary and sufficient condition of polynomial approximation type for the existence of universal elements. This certainly covers the case of simultaneous approximation with a sequence of continuous linear mappings. In the case of a sequence of unbounded operators the same condition ensures existence and density of universal elements. Several known results, stronger statements or new results can be...

Firstly, we give extensions of results of Gelfand, Esterle and Katznelson--Tzafriri on power-bounded operators. Secondly, some results and questions relating to power-bounded elements in the unitization of a commutative radical Banach algebra are discussed.

Cet exposé est une introduction au calcul étranger d’Écalle, c’est-à-dire au calcul des obstructions à la sommabilité de Borel d’une grande classe de séries formelles, les fonctions résurgentes d’Écalle. La théorie d’Écalle éclaire d’un jour neuf le célèbre phénomène de Stokes qui est illustré ici dans le contexte de la méthode du col.

We present a new method of analytic continuation of series out of their disk of convergence. We then exhibit a connection with the phenomenon of bifurcation delay in a planar discrete dynamical system; the limit of the method is then related to a stop phenomenon.

Soient $P\left(\underset{\_}{x}\right)=P(x_1,...,x_n)$ et $Q\left(\underset{\_}{x}\right)=Q(x_1,...,x_n)$ deux polynômes à coefficients positifs vérifiant :$$\underset{\genfrac{}{}{0pt}{}{\left|\underset{\_}{x}\right|\to +\infty }{x_1,...,x_n\ge 1}}{lim}\frac{P\left(\underset{\_}{x}\right)}{Q\left(\underset{\_}{x}\right)}=+\infty .$$Soient $\underset{\_}{\eta }=({\eta }_1,...,{\eta }_n)\in {\mathbf{N}}^n$ et $R=P/Q$. On étudie la série de Dirichlet $Z(R,\underset{\_}{\eta };s)={\sum }_{{\eta }_1,...,{\eta }_n=1}^{\infty }{\underset{\_}{\eta }}^{\underset{\_}{\eta }}R(\underset{\_}{\eta }{)}^{-s}$ : abscisse de convergence absolue, existence et nature du prolongement méromorphe, ordre de grandeur dans les bandes verticales. On donne un procédé de construction du prolongement méromorphe de la fonction $s\mapsto Z(R,\underset{\_}{\eta };s)$ qui ne dépend que de $\underset{\_}{\eta }$ et de certains monômes de $P$ et $Q$: les monômes extrémaux.