Reproducing Kernels and Conformal Mappings in Rn

In this paper we look at the theory of reproducing kernels for spaces of functions in a Clifford algebra R0, n. A first result is that reproducing kernels of this kind are solutions to a minimum problem, which is a non-trivial extension of the analogous property for real and complex valued functions. In the next sections we restrict our attention to Szeg¨o and Bergman modules of monogenic functions. The transformation property of the Szeg¨o kernel under conformal transformations is proved, and the Szeg¨o and Bergman kernels for the half space are calculated.