We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s, u) = ∑ ann −s−ū, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hil...

We consider the reproducing kernel Hilbert space Hμ induced by a kernel which is obtained using the Fourier-Stieltjes transform of a regular, positive, finite Borel measure μ on a locally compact abelian topological group Γ. Denote by G the dual of Γ. We determine Hμ as a certain subspace of the space C0(G) of all continuous function on G vanishing at infinity. Our main application is calculati...

Summary The performance of adaptive estimators that employ embedding in reproducing kernel Hilbert spaces (RKHS) depends on the choice location basis centers. Parameter convergence and error approximation rates depend where how centers are distributed state‐space. In this article, we develop theory relates parameter to position We criteria for choosing a specific class systems by exploiting fac...

Performance of the linear models, widely used within the framework of adaptive line enhancement (ALE), deteriorates dramatically in the presence of non-Gaussian noises. On the other hand, adaptive implementation of nonlinear models, e.g. the Volterra filters, suffers from the severe problems of large number of parameters and slow convergence. Nonetheless, kernel methods are emerging solutions t...

Data-Driven Optimization Using Reproducing Kernel Hilbert Spaces

where (Ω,μ) is a probability space. The kernel K is used to generate a Hilbert space, known as a reproducing kernel Hilbert space, whose unit ball is the class of functions we investigate. Recall that if K is a positive definite function K : Ω×Ω → R, then by Mercer’s Theorem there is an orthonormal basis (φi)i=1 of L2(μ) such that μ× μ almost surely, K(x,y) = ∑i=1 λiφi(x)φi(y), where (λi)i=1 is...

in this paper, a numerical scheme for solving singular initial/boundary value problems presented.by applying the reproducing kernel hilbert space method (rkhsm) for solving these problems,this method obtained to approximated solution. numerical examples are given to demonstrate theaccuracy of the present method. the result obtained by the method and the exact solution are foundto be in good agr...

We focus on three methods for finding a suitable subspace for regression in a reproducing kernel Hilbert space: kernel principal component analysis, kernel partial least squares and kernel canonical correlation analysis and we demonstrate how this fits within a more general context of subspace regression. For the kernel partial least squares case a least squares support vector machine style der...

This TR contains two brief reviews which will appear in the Proceedings of the 13th IFAC Symposium on System Identification (SYSID 2003), Rotterdam, August 2003. They are the basis for two talks in the invited session WeP02-Reproducing Kernels 1, and were prepared within the space limitations of the Proceedings. They are primarily based on work of the author and collaborators. There are many re...