Lagrange polynomials, reproducing kernels and cubature in two dimensions

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...

Cubature formulas, geometrical designs, reproducing kernels, and Markov operators

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to la...

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...

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...