Abel universal series

TL;DR

This paper introduces Abel universal series, a class of holomorphic functions with dense approximation properties on compact sets of the unit circle, and explores their unique characteristics and dynamical behaviors.

Contribution

It establishes the distinctness of Abel universal series from classical universal functions and analyzes their invariance and dynamical properties under dilation and differentiation.

Findings

Abel universal series form a residual set in H(D).

They are not comparable to functions with dense Taylor polynomials.

They are not invariant under differentiation.

Abstract

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.