New results on universal Taylor series via 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 $α$ K > 0 such that there exists a bounded domain G containing K such that the weighted polynomials of the form z $α$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 $α$ < $α$ K . Explicit computations of $α$ K are given for some simple compact sets K.