Quantitative incomplete polynomial approximation and frequently universal Taylor series

2025/04/28 by Stéphane Charpentier, Charpentier, Stéphane, Konstantinos Maronikolakis +1 · 1 citation
Computer Science · #30E10 #30K15 #41A10 #41A25 #47A16 #Complex Variables (math.CV) #FOS: Mathematics #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2504.20240

openalex publication_date 2025/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (τ_n)_n be a sequence of real numbers in (1,+∞). Using potential theoretic methods, we prove quantitative results - Bernstein-Walsh type theorems - about uniform approximation by polynomials of the form ∑_{k=\lfloor (n)/(τn) \rfloor}ⁿa_k z^k, on the union of two disjoint compact sets, one containing 0 and the other not. Moreover, we reveal the interplay between the compact sets and the asymptotic behaviour of the sequence (τ_n)_n. As applications of our results, we prove the existence of frequently universal Taylor series, with respect to the natural and the logarithmic densities, providing solutions to two problems posed by Mouze and Munnier.

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