Quantitative incomplete polynomial approximation and frequently universal Taylor series
Stéphane Charpentier, Konstantinos Maronikolakis
TL;DR
The paper develops a quantitative theory of incomplete polynomial approximation on pairs of disjoint compact sets via potential-theoretic methods, giving Bernstein–Walsh–type estimates for polynomials truncated at varying fractions of degree. It then applies these results to the theory of frequently universal Taylor series, proving the existence of log-density frequently universal Taylor series for any compact K in the complement of the unit disk and establishing density-type results for these universality properties. The work connects the geometry of the approximation sets, the asymptotics of the incompleteness parameter sequence (\tau_n), and potential-theoretic quantities such as capacity and Green functions to construct universal objects with respect to weighted densities, providing partial answers to questions posed by Mouze and Munnier. Overall, it advances both quantitative approximation theory and the study of universality in analytic function spaces, with explicit constructions and several open problems on extending the results to broader classes of compact sets and densities.
Abstract
Let $(τ_n)_n$ be a sequence of real numbers in $(1,+\infty)$. Using potential theoretic methods, we prove quantitative results - Bernstein-Walsh type theorems - about uniform approximation by polynomials of the form $\sum_{k=\lfloor \frac{n}{τ_n} \rfloor}^na_k z^k$, 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.