Universal Taylor series on specific compact sets

2015/06/04 by Nikos Tsirivas, Tsirivas, Nikos
Mathematics · #30B10Z #30E10 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.CV #msc:30B10Z #msc:30E10

paper · pdf · doi:10.48550/arxiv.1506.01528

arxiv created 2015/06/04 · openalex publication_date 2015/06/04 · arxiv updated 2015/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let D be the open unit disc in the complex plane. We denote by 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→ 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 2ⁿS_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 2ⁿS_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.

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