Abstract
We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.
Highlights
Many researchers have made use of reproducing kernels in attacking diverse areas and problems from approximation and optimization; see [1, 6, 15, 17, 19, 20]
While there is a direct link from specific positive definite kernels, the link to the corresponding reproducing kernel Hilbert space (RKHS) is a rather abstract one, typically, one is faced only with an abstract completion, and the links to computation is often blurred
It is well known that one can associate to k a uniquely defined Hilbert space (which we will denote by H(k)) of functions defined on S with reproducing kernel k(t, s), meaning that for every s ∈ S and f ∈ H(k), the function t → k(t, s) belong to H(k) and
Summary
Many researchers have made use of reproducing kernels in attacking diverse areas and problems from approximation and optimization; see [1, 6, 15, 17, 19, 20]. This will make use of the following five steps: (i) the filter of all finite subsets of S, (ii) an explicit system of finite-rank operators defined on functions on S, and (iii) an algorithm which related the operators defined from two finite sets, with one contained in the other.
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