Abstract
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 \(\mathbb{S}^{2}\) 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 \(\mathbf{i}\in \mathbb{S}^{2}\), are holomorphic maps. One of their crucial properties is that from the knowledge of the values of f on Ω∩ℂ(i) for some \(\mathbf{i}\in \mathbb{S}^{2}\), 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 \(\mathbf{i}\in \mathbb{S}^{2}\).
M.E. Luna-Elizarrarás and M. Shapiro were partially supported by CONACYT projects as well as by Instituto Politécnico Nacional in the framework of COFAA and SIP programs.
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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: Gentili, G., Sabadini, I., Shapiro, M., Sommen, F., Struppa, D. (eds) Advances in Hypercomplex Analysis. Springer INdAM Series, vol 1. Springer, Milano. https://doi.org/10.1007/978-88-470-2445-8_3
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