Dolomites Research Notes on Approximation

In this paper we consider the problem of approximating vector-valued functions over a domain Ω. 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 of the scalar-valued case. These spaces seem promising, when modelling correlations between the target function components, as the components are not learned independently of each other. 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.