Two universality results for polynomial reproducing kernels

Reproducing kernels , Bergman kernels , Poisson kernels

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...

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...

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...

Lagrange polynomials, reproducing kernels and cubature in two dimensions

We obtain by elementary methods necessary and sufficient conditions for a k-dimensional cubature formula to hold for all polynomials of degree up to 2m− 1 when the nodes of the formula have Lagrange polynomials of degree at most m. The main condition is that the Lagrange polynomial at each node is a scalar multiple of the reproducing kernel of degree m− 1 evaluated at the node plus an orthogona...