Partially smooth universal Taylor series on products of simply connected domains
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Partially smooth universal Taylor series on products of simply connected domains Giorgos Kotsovolis1 Received: 17 February 2020 / Accepted: 14 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Using a recent Mergelyan type theorem, we show the existence of universal Taylor series don products of planar simply connected domains i that extend continuously (i ∪ Si ), where Si are subsets of ∂ i , open in the relative topology. The on i=1 universal approximation occurs on every product of compact sets K i such that C − K i are connected and for some i 0 it holds K i0 ∩ (i0 ∪ Si0 ) = ∅. Keywords Universal Taylor series · Mergelyan’s theorem · Product of planar sets · Baire’s theorem · Abstract theory of universal series · Generic property Mathematics Subject Classification 30K05 · 32A05 · 32A17 · 32A30
1 Introduction In one complex variable, Mergelyan’s Theorem [17] has allowed the study of universal Taylor series [2,6,7,10–12,15,16]. However, the study of universal Taylor series in several complex variables is underdeveloped [3,4]. dRecently, [5] gave birth to a new K i , where K i are planar comMergelyan’s type theorem. In particular, if K = i=1 pact sets with connected complements and f : K → C is a continuous function such that f ◦ is holomorphic on the disk D ⊂ C, for every injective holomorphic mapping : D → K , then f is uniformly approximable on K by polynomials. This result wasrecently used in [9], where it is proved that there exist holomorphic functions d i , where i are simply connected domains of C, that behave universally on i=1 on all products K of planar compact sets, K i , i = 1, . . . , d, with connected comple-
Communicated by Adrian Constantin.
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Giorgos Kotsovolis [email protected] Department of Mathematics, National and Kapodistrian University of Athens, Panepistemiopolis, 157-84 Athens, Greece
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d ments C − K i , where K is disjoint from i=1 i . It is also proved that there exist d smooth functions on i=1 i , where i are simply connected domains of C, such that C − i is connected, which behave universally on all products K of planar compact sets, d K i , i = 1, . . . , d, with connected complements C − K i , where K is disjoint from these results are improved by proving the existence of i=1 i . In the present paper, d holomorphic functions on i=1 i , where i are planar simply connected domains d that extend smoothly on i=1 (i ∪ Si ), where Si are subsets of ∂ i , open in the relative topology of ∂ Si and C − (i ∪ Si ) is connected. The universal approximation is achieved on all products K of planar compact setsK i , i= 1, . . . , d, with connected d i ∪ S i . Partially smooth complements C − K i , where K is disjoint from i=1 universal Taylor series are not so developed even in one complex variable [8,19]. Our d K i , K i ⊂ C, C − K i method cannot give the result for all compact sets K = i=1 d connected, which are disjoint from i=1 (i ∪ Si ). In fact, we prove that there is no absorbing sequence of
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