Extension results for slice regular functions of a quaternionic variable

By Fabrizio Colombo, Graziano Gentili, Irene Sabadini, Daniele C. Struppa. 2009

141 citations recorded by the source corpus.

In this paper we prove a new Representation Formula for slice regular functions, which shows that the value of a slice regular function f at a point q=x+yI can be recovered by the values of f at the points q+yJ and q+yK for any choice of imaginary units I,J,K. This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/bf8dd00fdcff1edb50868feb95846b2b6faf06c1

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In Applied Mathematics

Citations in the corpus

Regular Functions on the Space of Cayley Numbers (2010, background): “A description of this theory can be found in [11], [12], but the interested reader can find the most recent developments in [10] and [ 3 ].”
The Fueter mapping theorem in integral form and the ℱ‐functional calculus (2010, background): “This class of functions has been introduced in [12] and further investigated in [13].”
A new series expansion for slice regular functions (2011, background): “Indeed, the Representation Formula proven in [3] can be restated as follows.”
On Two Approaches to the Bergman Theory for Slice Regular Functions (2013, methodology): “In this paper we introduce the first elements and the settings of the Bergman theory in the case of slice regular functions studied in the papers [5–7, 9, 17].”
Slice regular functions on real alternative algebras (2010, methodology): “Representation formulas for quaternionic Cullen regular functions appeared in [3] and [4].”
Twistor transforms of quaternionic functions and orthogonal complex structures (2012, methodology): “Recently, [4] identified a larger class of domains on which the theory of regular functions is natural and not limited to quaternionic power series.”
The inverse Fueter mapping theorem (2011, methodology): “The quaternionic version of the theory is based on the notion of slice regularity which was inspired by the paper of Cullen [16] and its further advances are in the papers [4], [18].”
Regular Functions of a Quaternionic Variable (2013, methodology): “The following definition of slice regular function was given in [5].”
Regular Moebius transformations of the space of quaternions (2008, background): “We now run through the basic properties of the zeros and the singularities of regular functions on B(0, R). Since the present paper is only concerned with regular functions on balls B(0, R) (including H = B(0, +∞)), we follow the presentation of [14,15,26], even though the algebraic structure has al”
Entire slice regular functions (2015, background): “The theory of slice regular functions started in [109] and [110] for functions defined at ball centered at the origin and then evolved in a series of papers, among which we mention [58], [46], [47], into a theory on axially symmetric domains.”
Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions (2012, unknown): “Slice hyperholomorphic functions have two main formulations according to the fact that the functions we consider are defined on quaternions and are quaternion-valued, in this case the functions are called slice regular, see [32, 15, 19] or the functions are defined on the Euclidean space R and have ”