Krein reproducing kernel modules in Clifford analysis

Abstract

Classic hypercomplex analysis is intimately linked with elliptic operators, such as the Laplacian or the Dirac operator, and positive quadratic forms. But there are many applications like the crystallographic X-ray transform or the ultrahyperbolic Dirac operator which are closely connected with indefinite quadratic forms. Although appearing in many papers in such cases Hilbert modules are not the right choice as function spaces since they do not reflect the induced geometry. In this paper we are going to show that Clifford–Krein modules are naturally appearing in this context. Even taking into account the difficulties, e.g., the existence of different inner products for duality and topology, we are going to demonstrate how one can work with them. Taking into account possible applications and the nature of hypercomplex analysis, special attention will be given to the study of Clifford–Krein modules with reproducing kernels. In the end we will discuss the interpolation problem in Clifford–Krein modules with reproducing kernel.