20170909Clifford-Krein modules with reproducing kernels
摘要:One of the central topics in hypercomplex function theory is interpolation. Interpolation by inner spherical monogenics provides the adequate counterpart to the classic finite Fourier transform and is widely used in applications. In more general terms, interpolation in hypercomplex function theory is closely linked to interpolation in reproducing kernel Hilbert spaces. But interestingly enough, in difference to the complex case there are many applications of hypercomplex function theory in which Hilbert spaces are not the natural setting. For instance, studying null-solutions of ultra-hyperbolic Dirac operators, pole figures of the crystallographic Radon transform which belong to the kernel of an ultra-hyperbolic Laplacian, or even more general problems in Minkowski space-time and string theory require inner product spaces with an indefinite inner product. Among other things this is due to the possibility of the underlying Clifford algebra to have a signature $(p,q)$ and, therefore, to be linked to Pontryagin modules instead of Hilbert modules. As we shall see the natural setting for such problems is the Clifford-Krein module. In this talk we give a general overview over Clifford-Krein modules with reproducing kernels and we are going to discuss interpolation in these modules.