Pontryagin space structure in reproducing kernel Hilbert spaces over *-semigroups

Reproducing kernel almost Pontryagin spaces

Article history: Received 16 May 2014 Accepted 1 August 2014 Submitted by R. Brualdi MSC: primary 46C20, 46E40 secondary 46E22, 54D35

A structure theorem for reproducing kernel Pontryagin spaces

We illustrate a relationship between reproducing kernel spaces and orthogonal polynomials via a general structure theorem. The Christofell–Darboux formula emerges as a limit case. c © 1998 Elsevier Science B.V. All rights reserved.

Real reproducing kernel Hilbert spaces

P (α) = C(α, F (x, y)) = αF (x, x) + 2αF (x, y) + F (x, y)F (y, y), which is ≥ 0. In the case F (x, x) = 0, the fact that P ≥ 0 implies that F (x, y) = 0. In the case F (x, y) 6= 0, P (α) is a quadratic polynomial and because P ≥ 0 it follows that the discriminant of P is ≤ 0: 4F (x, y) − 4 · F (x, x) · F (x, y)F (y, y) ≤ 0. That is, F (x, y) ≤ F (x, y)F (x, x)F (y, y), and this implies that F ...

Online learning in Reproducing Kernel Hilbert Spaces

Distribution Embeddings in Reproducing Kernel Hilbert Spaces

The “kernel trick” is well established as a means of constructing nonlinear algorithms from linear ones, by transferring the linear algorithms to a high dimensional feature space: specifically, a reproducing kernel Hilbert space (RKHS). Recently, it has become clear that a potentially more far reaching use of kernels is as a linear way of dealing with higher order statistics, by embedding proba...