Boundary behaviour of functions which possess universal Taylor series

Abstract

It 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 establishes a strong unboundedness property for such functions near every boundary point. The result is new even in the case of the disc, where it strengthens work of several authors.

Universal Taylor series, conformal mappings and boundary behaviour

“…Another question concerns boundary behaviour, about which there is a growing literature [18], [9], [13], [14], [7], [16], [1], [5], [11]. For example, in the case of the unit disc D, it is known that if f ∈ U(D, 0), then f does not belong to the Nevanlinna class (see [15]) and there is a residual subset Z of the unit circle T such that the set {f (rζ) : 0 < r < 1} is unbounded for every ζ ∈ Z (see [4]).…”

Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

“…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.…”

“…Further, Müller, Vlachou and Yavrian [15] showed that the collection U( ; ) is independent of the centre of expansion , from which it follows that no function f in U( ; ) can be extended holomorphically beyond (see also Theorem 8.4 in Melas and Nestoridis [14]). Recently, this conclusion was substantially strengthened in [9], where it was shown that each such function f is unbounded near every 0 2010 Mathematics Subject Classi…cation 30K05, 30B30, 30E10, 31A05. Keywords: holomorphic functions; universal approximation; Taylor series; Laurent series; boundary behaviour; subharmonic functions.…”

“…Since A c 0 is simply connected, Theorem 1 of [9] tells us that log + jf 0 j cannot have a harmonic majorant on D \ . The same must therefore also be true of log + jf j.…”

“…The hypotheses ensure that is multiply connected, by [9], and all components of c are bounded, by [15]. Without loss of generality we may assume that = 0 and < R. (In the general case we could then use the removability of polar sets for bounded holomorphic functions to see that f has a holomorphic continuation to D( ; )nD(0; R), and could apply the original case with replaced by an arbitrary point of @D(0; R) \ @ \ D( ; ).)…”