We are interested in approximating vector-valued functions on a compact
set $ØmegaR^d$. We consider reproducing kernel Hilbert
spaces of $R^m$-valued functions which each admit a unique
matrix-valued reproducing kernel $k$. These spaces seem promising,
when modelling correlations between the target function components.
The approximation of a function is a linear combination of matrix-valued
kernel evaluations multiplied with coefficient vectors. To guarantee
a fast evaluation of the approximant the expansion size, i.e. the
number of centers $n$ is desired to be small. We thus present three
different greedy algorithms by which a suitable set of centers is
chosen in an incremental fashion: First, the $P$-Greedy which requires
no function evaluations, second and third, the $f$-Greedy and $f/P$-Greedy
which require function evaluations but produce centers tailored to
the target function. The efficiency of the approaches is investigated
on some data from an artificial model.