We investigate VH-spaces (Vector Hilbert spaces, or Loynes spaces) operator valued Hermitian kernels that are invariant under actions of ∗-semigroups from the point of view of generation of ∗-representations, linearizations (Kolmogorov decompositions), and reproducing kernel spaces. We obtain a general dilation theorem in both Kolmogorov and reproducing kernel space representations, that unifie...

A resistance network is a connected graph (G, c) with edges (and edge weights) determined by the conductance function cxy . The Dirichlet energy form E produces a Hilbert space structure (which we call the energy space HE) on the space of functions of finite energy. In a previous paper, we constructed a reproducing kernel {vx} for this Hilbert space and used it to prove a discrete Gauss-Green i...

We formulate the problem of graph inference where part of the graph is known as a supervised learning problem, and propose an algorithm to solve it. The method involves the learning of a mapping of the vertices to a Euclidean space where the graph is easy to infer, and can be formulated as an optimization problem in a reproducing kernel Hilbert space. We report encouraging results on the proble...

A reproducing-kernel Hilbert space approach to image interpolation is introduced. In particular, the reproducing kernels of Sobolev spaces are shown to be exponential functions. These functions, in turn, give rise to alternative interpolation kernels that outperform presently available designs. Both theoretical and experimental results are presented.

Reproducing kernel Hilbert spaces and wavelets are both mathematical tools used in system identification and approximation. Reproducing kernel Hilbert spaces are function spaces possessing special characteristics that facilitate the search for solutions to norm minimization problems [3]. As such, they are of interest in a variety of areas including Machine Learning [11]. Wavelets are another mo...

We start by reviewing some elementary Banach and Hilbert space theory. Two key results here will prove useful in studying the properties of reproducing kernel Hilbert spaces: (a) that a linear operator on a Banach space is continuous if and only if it is bounded, and (b) that all continuous linear functionals on a Banach space arise from the inner product. The latter is often termed Riesz repre...

and Applied Analysis 3 Similarly, we define the space W 2 2 [0, T] = { { { { { { { V (t) | V (t) , V 󸀠 (t) are absolutely continuous in [0, T] , V 󸀠󸀠 (t) ∈ L 2 [0, T] , t ∈ [0, T] , V (0) = 0 } } }

We propose a method to learn simultaneously a vector-valued function and a kernel between its components. The obtained kernel can be used both to improve learning performance and to reveal structures in the output space which may be important in their own right. Our method is based on the solution of a suitable regularization problem over a reproducing kernel Hilbert space of vector-valued func...