Reproducing kernels and Beurling’s theorem

Favard theorem for reproducing kernels

Consider for n = 0, 1, . . . the nested spaces Ln of rational functions of degree n at most with given poles 1/αi, |αi| < 1, i = 1, . . . , n. Let L = ∪0 Ln. Given a finite positive measure μ on the unit circle, we associate with it an inner product on L by 〈f, g〉 = ∫ fgdμ. Suppose kn(z, w) is the reproducing kernel for Ln, i.e., 〈f(z), kn(z, w)〉 = f(w), for all f ∈ Ln, |w| < 1, then it is know...

Reproducing kernels , Bergman kernels , Poisson kernels

Duality by Reproducing Kernels

LetA be a determined or overdetermined elliptic differential operator on a smooth compact manifold X. Write A( ) for the space of solutions of the system Au= 0 in a domain X. Using reproducing kernels related to various Hilbert structures on subspaces of A( ), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of , we specify t...

Reproducing Kernels and Riccati Equations

The purpose of this article is to present a brief exposition of the role of Riccati equations in the theory of reproducing kernel spaces. In particular, we shall exhibit a connection between positive semidefinite solutions of matrix Riccati equations and a class of finite dimensional reproducing kernel Hilbert spaces of rational vector valued functions, and an analogous (but more general) conne...

Refinement of Reproducing Kernels

We continue our recent study on constructing a refinement kernel for a given kernel so that the reproducing kernel Hilbert space associated with the refinement kernel contains that with the original kernel as a subspace. To motivate this study, we first develop a refinement kernel method for learning, which gives an efficient algorithm for updating a learning predictor. Several characterization...