Universal Taylor series with maximal cluster sets

Abstract

We link the overconvergence properties of certain Taylor series in the unit disk to the maximality of their cluster sets, so connecting outer wild behavior to inner wild behavior. Specifically, it is proved the existence of a dense linear manifold of holomorphic functions in the disk that are, except for zero, universal Taylor series in the sense of Nestoridis and, simultaneously, have maximal cluster sets along many curves tending to the boundary. Moreover, it is constructed a dense linear manifold of universal Taylor series having, for each boundary point, limit zero along some path which is tangent to the corresponding radius. Finally, it is proved the existence of a closed infinite dimensional manifold of holomorphic functions enjoying the two-fold wild behavior specified at the beginning.

Universal Taylor series, conformal mappings and boundary behaviour

“…Another question concerns boundary behaviour, about which there is a growing literature [18], [9], [13], [14], [7], [16], [1], [5], [11]. For example, in the case of the unit disc D, it is known that if f ∈ U(D, 0), then f does not belong to the Nevanlinna class (see [15]) and there is a residual subset Z of the unit circle T such that the set {f (rζ) : 0 < r < 1} is unbounded for every ζ ∈ Z (see [4]).…”

Universal Taylor series and summability

“…In the present paper, we focus on the formulation of an intrinsic property of universal Taylor series which would guarantee that a universal Taylor series which enjoys this property has an image under some matrix summability method which is still universal (in the same sense as [8]). Furthermore, we prove that Questions (1) and (2) have a positive answer for a class of universal Taylor series on simply connected domains with large Ostrowski-gaps and, in passing, we formulate an answer to (3). No restriction on is required.…”

“…2). Several authors were recently interested in the dense-lineability and spaceability of certain kinds of universal Taylor series (see for instance [2,3,5,7,18] and the references therein). These concepts give some information about the algebraic structure of the sets of such series.…”

Linear subsets of nonlinear sets in topological vector spaces

“…In particular, some kind of wild behavior near the boundary (see Subsection 2.7.4) is compatible with the property of being a universal Taylor series. For instance, Bernal, Bonilla, Calderón, and Prado-Bassas [73] showed in 2009 that the family of universal Taylor series f having maximal (i.e., equal to C ∞ ) cluster set C(f, γ, ξ) at each ξ ∈ ∂D along any curve γ ⊂ D tending to ∂D whose closure does not contain ∂D is dense-lineable in H(D). Incidentally, in [80] it has been shown the maximal-dense-lineability in H(G) of the class of functions f ∈ U ((C ϕ n )) satisfying that boundary property, where G is a Jordan domain and (C ϕ n ), is the sequence of composition operators generated by adequate holomorphic self-mappings ϕ n : G → G.…”