Universal Taylor series for non-simply connected domains

Abstract

Stephen J. Gardiner and N. Tsirivas AbstractIt is known that, for any simply connected proper subdomain of the complex plane and any point in , there are holomorphic functions on that have "universal"Taylor series expansions about ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in Cn that have connected complement. This note shows that this phenomenon can break down for non-simply connected domains , even when Cn is compact. This answers a question of Melas and disproves a conjecture of Müller, Vlachou and Yavrian. RésuméIl est connu que, pour un sous-domaine propre simplement connexe du plan complexe et un point quelconque de , il y a des fonctions holomorphes sur qui possèdent des séries de Taylor « universelles» autour de ; c'est-à-dire tout polynôme peut être approximé, sur tout compact de Cn ayant un complémentaire connexe, par les sommes partielles de la série de Taylor. Cette note montre que ce résultat n'est plus vrai en général pour les domaines non-simplement connexes , même lorsque Cn est compact. Cela répond à une question de Melas et réfute une conjecture de Müller, Vlachou et Yavrian.

Boundary behaviour of functions which possess universal Taylor series

“…Further (cf. [8], page 250), since u k = log j j + g k for some subharmonic function g k on , and since G ( ; ) is the least non-negative superharmonic function on of the form log j j g, where g is subharmonic on , we see that…”

Determination of a Universal Series

“…However, if K is a compact connected set in C whose complement is also connected, then in Ω = C \ K there exist universal Taylor series with respect to one center [8], [24], [37]; see also [3], [5], [11], [13], [14], [15], [31], [35], [38]. In this section we present three new propositions in the doubly connected case Ω = C \ K and the proofs presented here do not use Baire's Theorem and, as in the previous section, if ∂K is good enough they can be transformed to be realized infinite denumerable procedure.…”

Existence of Universal Taylor Series for Nonsimply Connected Domains

“…The result of [10], that U ( ; 0) = ; when is a domain of the form Cn(L [ f1g) and L is a non-degenerate continuum in CnD, is clearly a particular case of Corollary 2. This corollary also allows us to write down an example of a domain with bounded complement that admits no holomorphic functions with universal Taylor series expansions about even a single point:…”

“…On the other hand, Müller, Vlachou and Yavrian [17] have shown, for non-simply connected domains , that thinness of the complementary set Cn at in…nity is necessary for U ( ; ) to be non-empty. They conjectured that this condition is also su¢ cient, but it has recently been shown [10] that U ( ; 0) = ; when is a domain of the form Cn(L [ f1g) and L is a non-degenerate continuum in CnD. Here D = D(0; 1) and D( ; r) = fz 2 C : jz j < rg.…”