Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains

Abstract

Abstract. A holomorphic function on a planar domain is said to possess a universal Taylor series about a point of if subsequences of the partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compact sets in Cn that have connected complement. In the case where is simply connected such functions are known to be unbounded, and to form a collection that is independent of the choice of . This paper uses tools from potential theory to show that, even for domains of arbitrary connectivity, such functions are unbounded whenever they exist. In the doubly connected case a further analysis of boundary behaviour reveals that the collection of such functions can depend on the choice of . This phenomenon was previously known only for domains that are at least triply connected. Related results are also established for universal Laurent series.

Abel universal series

“…The functions in U(D) were extensively studied from many points of view. We refer to [1,4,12,18,20,21,22,29,30,32] and the references therein for a non-exhaustive list of papers. In particular, some of these highlight the fact that functions in U(D) enjoy irregular boundary behaviour, for example along radii.…”

A convergence theorem for harmonic measures with applications to Taylor series

“…(It was stated there for f ∈ U (ω, ξ), but the proof is valid also for f ∈ U 0 (ω, ξ).) Theorem 1 of [6] tells us that, if ζ ∈ ∂D 0 \ω and a function f in U (ω, ξ) is bounded in D(ζ, ρ) ∩ ω for some ρ > 0, then C\(ω ∪ D 0 ) must be polar. Corollary 2 yields the additional information that (∂D 0 \ω) ∩ D(ζ, ρ) must have zero arc length measure.…”

Common hypercyclic vectors and universal functions

“…Significant work in this area has made by Stephen Gardiner and other researchers see [10], [11], [12], [13] and [14].…”