Universal Taylor series on specific compact sets

TL;DR

This paper investigates conditions under which scaled Taylor partial sums of holomorphic functions are dense in spaces of functions on certain compact sets, highlighting the importance of the geometry of these sets.

Contribution

It demonstrates the existence of holomorphic functions whose scaled Taylor partial sums are dense in function spaces on specific compact sets, emphasizing the role of set geometry.

Findings

Existence of functions with dense scaled Taylor partial sums on certain compact sets.

The geometry of the compact set K is crucial for density results.

Generalization of results to arbitrary simply connected domains.

Abstract

Let $D$ be the open unit disc in the complex plane. We denote by $\mathbb{C}$ the set of complex numbers and consider any compact set $K$ which is disjoint from $D$ and which also has connected complement. Let $A(K)$ denote all the functions $f:K\to \mathbb{C} $ such that $f$ is continuous on $K$ and holomorphic in $K^o$. It is well known that there exist holomorphic functions $f$ on $D$ for which the partial sums $S_n(f)$, n=1,2,... of the Taylor series with center $0$ are dense in $A(K)$ for every $K$ satisfying the properties above. It is also known that the above result fails if we consider the weighted polynomials $2^nS_n(f)$, n=1,2,... instead of $S_n(f)$, n=1,2,.... In the opposite direction, the main result of this work shows that there exist holomorphic functions $f$ on $D$ for which the sequence $2^nS_n(f)$, $n=1,2,...$ is dense in $A(K)$ for specific compact sets $K$. In this case the geometry of $K$ plays a crucial role. We also generalize these results on arbitrary simply connected domains.