Reproducing kernels and orthogonal kernels for analytic differentials on Riemann surfaces

Reproducing kernels , Bergman kernels , Poisson kernels

Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials p minimizing Dirichlet-type norms ‖pf − 1‖α for a given function f . For α ∈ [0, 1] (which includes the Hardy and Dirichlet spaces of the disk) and general f , we show that such extremal polynomials are non-vanishing in the closed unit disk. For negative α, the weighted Bergman space case, the ext...

Hypergeometric reproducing kernels and analytic continuation from a half-line

Indefinite inner product spaces of entire functions and functions analytic inside a disk are considered and their completeness studied. Spaces induced by the rotation invariant reproducing kernels in the form of the generalized hypergeometric function are completely characterized. A particular space generated by the modified Bessel function kernel is utilized to derive an analytic continuation ...

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...