arXiv subjects

A Mouze

Publications and source records attributed to A Mouze.

4 recordsLinked to original sources

arXiv2022

Abel universal series

Given a sequence = (r n) n $\in$ [0, 1) tending to 1, we consider the set U A (D,) of Abel universal series consisting of holomorphic functions f in the open unit disc D such that for any compact set K included in the unit circle T, different from T, the set {z $\rightarrow$ f (r n $\bullet$)| K : n $\in$ N} is dense in the space C(K) of continuous functions on K. It is known that the set U A (D,) is residual in H(D). We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at 0 are dense in C(K) for any compact set K $\subset$ T different from T. Moreover we prove that the class of Abel universal series is not invariant under the action of the differentiation operator. Finally an Abel universal series can be viewed as a universal vector of the sequence of dilation operators T n : f $\rightarrow$ f (r n $\bullet$) acting on H(D). Thus we study the dynamical properties of (T n) n such as the multi-universality and the (common) frequent universality. All the proofs are constructive.

math.CV↗

arXiv2018

Frequent universality criterion and densities

We improve a recent result by giving the optimal conclusion possible both to the frequent universality criterion and the frequent hypercyclicity criterion using the notion of A-densities, where A refers to some weighted densities sharper than the natural lower density. Moreover we construct an operator which is logarithmically-frequently hypercyclic but not frequently hypercyclic.

math.FA↗

arXiv2017

On doubly universal functions

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 $\sum\_{k\geq 0}a\_kz^k$ with radius of convergence 1 such that the pairs of partial sums $\{(\sum\_{k=0}^na\_kz^k,\sum\_{k=0}^{λ\_n}a\_kz^k): n=1,2,\dots\}$ approximate all pairs of polynomials uniformly on compact subsets $K\subset\{z\in\mathbb{C} :| z|\textgreater{}1\},$ with connected complement, if and only if $\limsup\_{n}\frac{λ\_n}{n}=+\infty.$ In the present paper, we give a new proof of this statement avoiding the use of advanced tools of potential theory. It allows to obtain the algebraic genericity of the set of such power series and to study the case of doubly universal infinitely differentiable functions. Further we show that the Cesàro means of partial sums of power series with radius of convergence 1 cannot be frequently universal.

math.CA↗

arXiv2016

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.

math.FA↗