Positive definite kernels play an increasingly prominent role in many applications such as scattered data fitting, numerical solution of PDEs, computer experiments, machine learning, rapid prototyping and computer graphics. We discuss some of the historical and current developments of the theory and applications of positive definite kernels – always with an eye toward the mathematics of Götting...

A new notion of stochastic germs for quantum processes is introduced and a characterisation of the stochastic differentials for positive definite (PD) processes is found in terms of their germs for arbitrary Itô algebra. A representation theorem, giving the pseudo-Hilbert dilation for the germ-matrix of the differential, is proved. This suggests the general form of quantum stochastic evolution ...

Article history: Received 13 November 2014 Accepted 25 August 2015 Available online xxxx Submitted by V. Mehrmann MSC: 15A15 15A39 15A69 46M05 47A63

This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as reproducing kernel Hibert spaces, the natural extension of the set of functions {k(x, ·), x ∈ X} associated with a kernel k defined on a space X . We discuss at le...