Function Theory for Finite Difference Dirac Operators

摘要

The complex function theory as well as the classical potential theory and the function theory of quaternion-valued functions can be understood as differnet models in the framework of the algebraic theory of right invertible operators (see e.g. (1)). Using one-dimensional examples we show that also numerical methods like the finite element method and the finite difference method will be covered by this approach. Furthermore the abstract interpolation theory, best approximation methods, boundary element methods, and projection methods will be related to the theory of right invertible operators. These considerations will be used to discuss first verisions of a discrete complex function theory from a new point of view. Finally, a function theory for several finite difference Dirac operators will be developed. Different models will be compared and applied to a boundary value problem.