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The Bergman–Sce transform for slice monogenic functions
Author(s) -
Colombo Fabrizio,
Gonzàles Cervantes Jose O.,
Sabadini Irene
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1489
Subject(s) - bergman kernel , mathematics , kernel (algebra) , complex plane , pure mathematics , bergman space , mathematical analysis , algebra over a field , bounded function
In a recent paper, we showed that the classical Bergman theory admits two possible formulations for the class of slice regular functions with quaternionic values. In the so called formulation of the first kind , we provide a Bergman kernel which is defined on U ⊂ H and is a reproducing kernel. In the so called formulation of the second kind , we use the Representation Formula for slice regular functions to define a second Bergman kernel; this time the kernel is still defined on U , but the integral representation of f is based on an integral computed only on U ∩ C Iand the integral does not depend on I ∈ S 2 , (here S denotes the sphere unit of purely imaginary quaternions, andC Irepresents the complex plane with imaginary unit I ). In this paper, we extend the second formulation of the Bergman theory to the case of slice monogenic functions and we focus our attention on the so‐called Bergman–Sce transform. This integral transform is defined by using the Bergman kernel and the Sce mapping theorem and associates to every slice monogenic function f , an axially monogenic functionf ̆ . Copyright © 2011 John Wiley & Sons, Ltd.