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Holomorphic functions with universal boundary behaviour

Résumé

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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Dates et versions

hal-01822216 , version 1 (24-06-2018)

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S. Charpentier. Holomorphic functions with universal boundary behaviour. 2018. ⟨hal-01822216⟩
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