  In this paper we consider the problem of approximating vector-valued
functions over a domain $\Omega$. For this purpose, we use matrix-valued
reproducing kernels, which can be related to Reproducing kernel Hilbert spaces
of vectorial functions and which can be viewed as an extension to the
scalar-valued case. These spaces seem promising, when modelling correlations
between the target function components, as the components are not learned
independently of one another. We focus on the interpolation with such
matrix-valued kernels. We derive error bounds for the interpolation error in
terms of a generalized power-function and we introduce a subclass of
matrix-valued kernels whose power-functions can be traced back to the
power-function of scalar-valued reproducing kernels. Finally, we apply these
kind of kernels to some artificial data to illustrate the benefit of
interpolation with matrix-valued kernels in comparison to a componentwise
approach.
