In this paper we show that the classical Bergman theory admits two possible settings for the class of slice regular functions. Let Ω be a suitable open subset of the space of quaternions ℍ that intersects the real line and let 𝕊2 be the unit sphere of purely imaginary quaternions. Slice regular functions are those functions f: Ω →H whose restriction to the complex planes ℂ(i), for every i ∈ S2, are holomorphic maps. One of their crucial properties is that from the knowledge of the values of f on Ω ∩ ℂ(i) for some i ∈ 𝕊2, one can reconstruct f on the whole Ω by the so called Representation Formula. We will define the so-called slice regular Bergman theory of the first kind. By the Riesz representation theorem we provide a Bergman kernel which is defined on Ω and is a reproducing kernel. In the slice regular Bergman theory of the second kind we use the Representation Formula to define another Bergman kernel; this time the kernel is still defined on Ω but the integral representation of f requires the calculation of the integral only on Ω ∩ℂ(i) and the integral does not depend on i ∈ 𝕊2.
CITATION STYLE
Colombo, F., González-Cervantes, J. O., Luna-Elizarrarás, M. E., Sabadini, I., & Shapiro, M. (2013). On two approaches to the Bergman theory for slice regular functions. In Springer INdAM Series (Vol. 1, pp. 39–54). Springer International Publishing. https://doi.org/10.1007/978-88-470-2445-8_3
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