- Gardiner, Stephen J.

###### Options

# Gardiner, Stephen J.

Preferred name

Gardiner, Stephen J.

Official Name

Gardiner, Stephen J.

## Research Output

17 results Back to results

### Filters

##### Author

##### Subject

##### Has files

##### Type

### Settings

Sort By

Results per page

Now showing 1 - 10 of 17

- PublicationRecent progress on fine differentiability and fine harmonicityThis paper describes recent results concerning the notions of differentiability and harmonicity with respect to the ne topology of classical potential theory.
338 - PublicationUniversal Taylor series, conformal mappings and boundary behaviourA holomorphic function f on a simply connected domain Ω is said to possess a universal Taylor series about a point in Ω if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta K outside Ω (provided only that K has connected complement). This paper shows that this property is not conformally invariant, and, in the case where Ω is the unit disc, that such functions have extreme angular boundary behaviour.
264 - PublicationBoundary Behaviour of Universal Taylor Series on Multiply Connected DomainsA holomorphic function on a planar domain Ω is said to possess a universal Taylor series about a point ζ of Ω if subsequences of the partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compact sets in C∖Ω that have connected complement. In the case where Ω is simply connected, such functions are known to be unbounded and to form a collection that is independent of the choice of ζ. This paper uses tools from potential theory to show that, even for domains Ω 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 ζ. This phenomenon was previously known only for domains that are at least triply connected. Related results are also established for universal Laurent series.
372Scopus© Citations 5 - PublicationHarmonic functions which vanish on a cylindrical surfaceSuppose that a harmonic function h on a finite cylinder vanishes on the curved part of the boundary. This paper answers a question of Khavinson by showing that h then has a harmonic continuation to the infinite strip bounded by the hyperplanes containing the flat parts of the boundary. The existence of this extension is established by an analysis of the convergence properties of a double series expansion of the Green function of an infinite cylinder beyond the domain itself.
251Scopus© Citations 9 - PublicationBoundary behaviour of Dirichlet series with applications to universal seriesThis paper establishes connections between the boundary behaviour of functions representable as absolutely convergent Dirichlet series in a half-plane and the convergence properties of partial sums of the Dirichlet series on the boundary. This yields insights into the boundary behaviour of Dirichlet series and Taylor series which have universal approximation properties.
267Scopus© Citations 6 - PublicationExtension results for harmonic functions which vanish on cylindrical surfacesThe Schwarz reflection principle applies to a harmonic function which continuously vanishes on a relatively open subset of a planar or spherical boundary surface. It yields a harmonic extension to a predefined larger domain and provides a simple formula for this extension. Although such a point-to-point reflection law is unavailable for other types of surface in higher dimensions, it is natural to investigate whether similar harmonic extension results still hold. This article describes recent progress on such results for the particular case of cylindrical surfaces, and concludes with several open questions.
390Scopus© Citations 2 - PublicationStationary Boundary Points for a Laplacian Growth Problem in Higher DimensionsIt is known that corners of interior angle less than π/2 in the boundary of a plane domain are initially stationary for Hele–Shaw flow arising from an arbitrary injection point inside the domain. This paper establishes the corresponding result for Laplacian growth of domains in higher dimensions. The problem is treated in terms of evolving families of quadrature domains for subharmonic functions.
250 - PublicationA characterization of annular domains by quadrature identitiesThis note verifies a conjecture of Armitage and Goldstein that annular domains may be characterized as quadrature domains for harmonic functions with respect to a uniformly distributed measure on a sphere.
317Scopus© Citations 3 - PublicationA reflection result for harmonic functions which vanish on a cylindrical surfaceSuppose that a harmonic function h on a finite cylinder U vanishes on the curved part A of the boundary. It was recently shown that h then has a harmonic continuation to the infinite strip bounded by the hyperplanes containing the flat parts of the boundary. This paper examines what can be said if the above function h is merely harmonic near A (and inside U). It is shown that h then has a harmonic extension to a larger domain formed by radial reflection.
316Scopus© Citations 7 - PublicationExistence of universal Taylor series for non-simply connected domainsIt is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that possess “universal” Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in ℂ\Ω that have connected complement. This paper shows, for nonsimply connected domains Ω, how issues of capacity, thinness and topology affect the existence of holomorphic functions on Ω that have universal Taylor series expansions about a given point.
358Scopus© Citations 11