Given a positive denite kernel , there is an associated reproducing kernel Hilbert space with reproducing kernel . The fact that the solution of certain penalized approximation problems, e.g. the smoothing spline problem, is given as a linear combination of \point evaluations" of the kernel is commonly referred to as the representer theorem in machine learning. The talk will discuss the analog statement for a conditionally positive denite, matrix-valued kernel, where the associated space turns out to be a reproducing kernel Pontryagin space of vector-valued functions