Existence of Universal Taylor Series for Nonsimply Connected Domains

Abstract

Publication informationConstructive Approximation, 35 (2) Stephen J. Gardiner AbstractIt is known that, for any simply connected proper subdomain of the complex plane and any point in , there are holomorphic functions on that possess "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 paper shows, for non-simply connected domains , how issues of capacity, thinness and topology a¤ect the existence of holomorphic functions on that have universal Taylor series expansions about a given point.

Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

“…Further, if c nD(0; R) is non-polar, then we choose K so that KnD(0; R) is also non-polar. If c \ D( ; ) is polar, then f obviously has a holomorphic continuation to D( ; ); also, from Corollary 1 of [8] and the fact that U( ; 0) 6 = ;, we see that c nD(0; R) must be polar. We may therefore assume from now on that c \ D( ; ) is non-polar.…”

“…The result below is a variant of Theorem 2 in [8], which covered the case where c is a disc. We will give a substantially new proof.…”

“…and z. Finally, we denote the characteristic function of a set A by A , and write v for the upper (respectively, b v for the lower) semicontinuous regularization of a function v. We begin by recalling two results, which can be found in [8] and [9], respectively. Theorem A.…”

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.…”

Πολλαπλά Καθολικές Συναρτήσεις

“…Πολλοί ερευνητές έχουν εργασθεί στην περιοχή των καθολικών σειρών και υπάρχει πληθώρα αξιόλογων άρθρων. Πέρα από τα άρθρα που έχουμε ήδη αναφέρει, παραπέμπουμε ενδεικτικά στα ακόλουθα άρθρα: [43], [47], [27], [29], [44], [30], [59], [40], [46], [60], [24]. Παραπέμπουμε επίσης στο άρθρο [32] του K.G.…”