  This work concerns the construction and characterization of product kernels
for multivariate approximation from a finite set of discrete samples. To this
end, we consider composing different component kernels, each acting on a
low-dimensional Euclidean space. Due to Aronszajn (1950), the product of
positive semi-definite kernel functions is again positive semi-definite, where,
moreover, the corresponding native space is a particular instance of a tensor
product, referred to as Hilbert tensor product. We first analyze the general
problem of multivariate interpolation by product kernels. Then, we further
investigate the tensor product structure, in particular for grid-like samples.
We use this case to show that the product of strictly positive definite kernel
functions is again strictly positive definite. Moreover, we develop an
efficient computation scheme for the well-known Newton basis. Supporting
numerical examples show the good performance of product kernels, especially for
their flexibility.
