Konstruktion, Approximation und Eigenschaften hyperkomplexer Kernfunktionen und ihre Anwendung auf partielle Differentialgleichungen

The present thesis is set within the field of hypercomplex function theory, which regards the funtion theory over Clifford algebras. The central term on which this thesis revolves is the reproducing kernel of a Hilbert function space. The thesis treats a number of different questions, which all involve two special cases of a reproducing kernel function, the so-called Bergman kernel and the so-called Szegö kernel. Firstly, the basic definitions as well as important and needed results from hypercomplex function theory are recalled. Secondly, the existence and construction of said reproducing kernel functions is treated for certain classes of domains. The obtained representations of the kernel functions are then used to solve certain partial differential equations. Furthermore, solutions to certain classes of Dirichlet problems are given using kernel functions and corresponding differential operators. The next item of research is from the field of differential geometry. A metric is defined by using the Szegö kernel, and its properties concerning curvature and completeness are verified. This is done in view of the connection of a similarly defined metric in the context of function theory in several complex variables and the smoothness of the underlying domain. Finally, the field of number theory is examined. For a start, a long standing problem within hypercomplex function theory is solved, as the existence and construction of non-trivial cusp forms can be verified and respectively performed. Furthermore, by using the Bergman kernel, a generalization of the Selberg trace formula for this newly constructed class of functions is proposed and proven