New results on universal Taylor series via weighted polynomial approximation

TL;DR

This paper advances the theory of universal Taylor series by employing weighted polynomial approximation to establish new density results and construct functions with specific growth properties of Taylor coefficients.

Contribution

It introduces novel weighted polynomial approximation techniques to prove the existence of dense sets of functions and constructs holomorphic functions with unbounded Taylor coefficient moduli.

Findings

Existence of a compact set K with dense holomorphic functions in A(K)

Construction of holomorphic functions with Taylor coefficients tending to infinity

Explicit computation of critical constants for weighted polynomial approximation

Abstract

We use weighted polynomial approximation to prove the existence of a compact set K with non-empty interior and a function f is dense in the space A(K) of all continuous functions on K that are holomorphic in the interior of K, endowed with the sup norm, while the set This improves a result of Mouze. The main ideas of the proof also allows us to construct a holomorphic function while the modulus of its non-zero Taylor coecients go to $\infty$. In passing, we complement a result by Pritsker and Varga on weighted polynomial approximation by proving that, for any compact set K with connected complement, there exists a constant $\alpha$ K > 0 such that there exists a bounded domain G containing K such that the weighted polynomials of the form z $\alpha$n P n , with deg(P n ) $\le$ n, are dense in H(G) for the topology of locally uniform convergence if and only if $\alpha$ < $\alpha$ K . Explicit computations of $\alpha$ K are given for some simple compact sets K.