Simultaneously maximal radial cluster sets

Bernal–González, Luis; Calderón-Moreno, M. C.; Prado-Bassas, J. A.

doi:10.1016/j.jat.2005.04.002

Cited by 7 publications

(2 citation statements)

References 11 publications

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“…(See also [15,Theorem 2.1] for an extension of the Kierst-Szpilrajn statement where certain holomorphic operators participate.) Since the intersection of two residual subsets is also residual, we can assert, as a consequence of the results by Kierst-Szpilrajn and Nestoridis, that "most" functions in H(D) exhibit, simultaneously, an extremely wild "inner" and "outer" boundary behavior; in other words, this two-fold property is "topologically generic".…”

Section: A Holomorphic Function F ∈ H(d) Is Called a Universal Taylormentioning

confidence: 99%

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Universal Taylor series with maximal cluster sets

Bernal–González¹,

Bonilla²,

Calderón-Moreno³

et al. 2009

Rev. Mat. Iberoam.

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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.

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Section: A Holomorphic Function F ∈ H(d) Is Called a Universal Taylormentioning

confidence: 99%

“…The condition (15) ensures that the sequence {f j,k } j≥k converges uniformly on any compact subset of D to a function f k ∈ H(D). Let E be the vector space consisting of all series ∞ k=1 c k f k converging uniformly on compacta of D, and let F be the closure of E in H(D).…”

Section: Closed Linear Manifolds Of Wild Functionsmentioning

confidence: 99%

Universal Taylor series with maximal cluster sets

Bernal–González¹,

Bonilla²,

Calderón-Moreno³

et al. 2009

Rev. Mat. Iberoam.

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Large subspaces of compositionally universal functions with maximal cluster sets

Bernal–González

Calderón-Moreno

Prado-Bassas

2012

Journal of Approximation Theory

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Holomorphic functions with universal boundary behaviour

Charpentier

2020

Journal of Approximation Theory

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We are interested in functions analytic in the unit disc D of the complex plane C with a wild behaviour near the boundary T of D. For instance, the main result implies the existence of a residual subset of H(D) whose every element f satisfies the property that, given any compact subset K of T, different from T, given any continuous functions ϕ on K, and any compact set L of D, there exists an increasing sequence (r n) n ⊂ [0, 1) converging to 1, such that f (r n (ζ − z) + z) converges to ϕ(ζ), uniformly for (ζ, z) ∈ K × L, as n goes to ∞. Among other things radial growth of such functions and connections with universal Taylor series are investigated. Functions in the disc algebra whose derivatives disjointly satisfy a somewhat similar universal property, are also exhibited. In some sense, the results are sharp.

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Simultaneously maximal radial cluster sets

Cited by 7 publications

References 11 publications

Universal Taylor series with maximal cluster sets

Universal Taylor series with maximal cluster sets

Large subspaces of compositionally universal functions with maximal cluster sets

Holomorphic functions with universal boundary behaviour

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