UNIVERSAL TAYLOR SERIES WITH RESPECT TO A PRESCRIBED SUBSEQUENCE

Augustin Mouze (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

For a holomorphic function f in the open unit disc D and ζ ∈ D, Sn(f, ζ) denotes the n-th partial sum of the Taylor development of f at ζ. Given an increasing sequence of positive integers (µn), we consider the classes of such functions f such that the partial sums {Sn(f, ζ) : n = 1, 2,. .. } (resp. {Sµ n (f, ζ) : n = 1, 2,. .. }) approximate all polynomials uniformly on the compact sets K ⊂ {z ∈ C : |z| ≥ 1} with connected complement. We show that these two classes of universal Taylor series coincide if and only if lim sup n µ n+1 µn < +∞. Finally we establish a similar result for real universal Taylor series.

Domaines: Analyse classique [math.CA]; Variables complexes [math.CV]; Analyse fonctionnelle [math.FA]

Augustin Mouze. UNIVERSAL TAYLOR SERIES WITH RESPECT TO A PRESCRIBED SUBSEQUENCE. 2020. ⟨hal-02877778v1⟩